Totally Unimodular Congestion Games
Totally Unimodular Congestion Games Alberto Del Pia∗
Michael Ferris†
Carla Michini‡
arXiv:1511.02784v1 [cs.GT] 9 Nov 2015
November 10, 2015
Abstract We investigate a new class of congestion games, called Totally Unimodular Congestion Games, in which the strategies of each player are expressed as binary vectors lying in a polyhedron defined using a totally unimodular constraint matrix and an integer right-hand side. We study both the symmetric and the asymmetric variants of the game. In the symmetric variant, all players have the same strategy set, i.e. the same constraint matrix and right-hand side. Network congestion games are an example of this class. Fabrikant et al. proved that a pure Nash equilibrium of symmetric network congestion games can be found in strongly polynomial time, while the asymmetric network congestion games are PLS-complete. We give a strongly polynomial-time algorithm to find a pure Nash equilibrium of any symmetric totally unimodular congestion game. We also identify four totally unimodular congestion games, where the players’ strategy sets are matchings, vertex covers, edge covers and stable sets of a given bipartite graph. For these games we derive specialized combinatorial algorithms to find a pure Nash equilibrium in the symmetric variant, and show the asymmetric variant is PLS-complete.
1
Introduction
A central problem of Algorithmic Game Theory concerns the existence of Nash equilibria and the design of polynomial-time algorithms for their computation. Nash equilibria are a very powerful and well-studied solution concept, and they have had tremendous impact in economics and social sciences [9]. A mixed strategy is a probability distribution over the pure strategies of a player. In his celebrated work [12, 13], John Nash proved the existence of mixed equilibria for any game with a finite set of strategies. More recently it has been shown that finding a mixed equilibrium is PPAD-complete [6, 7]. In some applications mixed Nash equilibria seems to have no natural interpretation, and thus, pure Nash equilibria are sometimes a more suitable solution concept than mixed Nash equilibria [23, 1]. Unfortunately, pure Nash equilibria are not guaranteed to exist, and even if they do, they are often hard to compute. Thus, much effort has been devoted to designing polynomial-time algorithms to determine an approximate pure Nash equilibrium [3, 4, 5, 14]. On the positive side, there are classes of games that are known to posses pure Nash equilibria. A prominent example are potential games, originally introduced by Monderer and Shapley [11]. The key property of potential games is the existence of a potential function, i.e. a function defined on the joint strategy sets of the players, and such that, if any player unilaterally deviates from her strategy, the change in her payoff is equal to the change in the potential function. A pure Nash equilibrium of a potential game can be found with a local search algorithm that minimizes the potential function over the strategy space of the game, where each iteration of the algorithm corresponds to an improving step of a player. Even if this algorithm is finite, it could take an exponential number of iterations ∗ Department
of Industrial and Systems Engineering, University of Wisconsin-Madison. Email: [email protected] of Computer Sciences, University of Wisconsin-Madison. Email: [email protected] ‡ Wisconsin Institute for Discovery, University of Wisconsin-Madison. Email: [email protected]
† Department
1
to reach a local optimum, or equivalently, a pure Nash equilibrium. Potential games belong to the complexity class Polynomial Local Search (PLS), introduced by Johnson, Papadimitriou and Yannakakis [10, 18]. In fact, this class includes all problems with a local search algorithm where each improving step can be performed in polynomial time. Many families of potential games have been shown to be PLS-complete, suggesting that a polynomial-time algorithm to find a pure Nash equilibrium is unlikely to exist for these games in general. In this work, we focus on congestion games, a class of potential games that has been widely investigated in the literature [15, 17, 21, 8, 16, 4, 5], and we provide new insight on their computational complexity using a polyhedral approach. In a congestion game, a set of resources is given, and each player selects a feasible subset of the resources in order to minimize her cost function. The cost of a player’s strategy is the sum of the delays of the resources selected by the player, and the delay of each resource is a function of the total number of players using it. Feasible means that not all possible combinations of resources belong to the strategy set of a player. In particular, we will assume that the strategies of each player are implicitly given through a polyhedral representation. For example, we can define a congestion game where the set of resources is the node set of a bipartite graph G = (V, E), and the players’ strategies are the vertex covers of G. Figure 1 shows an instance of this game, where there are 3 players and each node has different delays when it is used by 1,2 or 3 players. If player 1 chooses {a, e, f }, player 2 chooses {c, d, e}, and player 3 chooses {d, e, f }, then their costs are 8, 9 and 10, respectively. However, this is not a pure Nash equilibrium, since player 1 could unilaterally deviate and choose vertex cover {a, b, c}, decreasing her cost by 1. Another example are network congestion games, where the resources are the arcs of a given digraph D = (V, E) and the strategies of each player i are all (ri , si )-paths in D, ri , si ∈ V . Fabrikant et al. [8] gave an algorithm to find a pure Nash equilibrium in symmetric network congestion games, i.e. when all players share the same origin-destination pair. The algorithm is based on a reduction to minimum cost flow and runs in strongly polynomial time. Fabrikant et al. also showed that asymmetric network congestion games are PLS-complete. Polyhedral approach. In a general congestion game with N players and n resources, let X i = {χxi : xi is a strategy of player i}, where χ denotes the incidence vector, and let P i = {xi ∈ [0, 1]n : Ai xi ≥ bi } be a polyhedron such that P i ∩ {0, 1}n = X i . We can formulate the problem of player i as Pn −i i minxi j=1 cj (x )xj (1) i i s.t. x ∈ P ∩ {0, 1}n , where x−i is the vector encoding the strategies of all the other players and cj (x−i ) is the cost that i would incur by selecting resource j, given x−i . Note that the objective of (1) is linear for fixed x−i . Our goal is to understand what properties of P i affect the computational complexity of finding a pure Nash equilibrium. Clearly, if X i is explicitly given for all i, we can find a pure Nash equilibrium by exhaustive search of the payoff matrix. In polyhedral terms, this would imply that, for each i and fixed x−i , we could solve (1) using the vertex description of conv(X i ). The complexity of this linear programming problem would indeed be polynomial in the input size. We first consider the class of symmetric congestion games, i.e. congestion games with Ai = A ∈ Rm×n and bi = b ∈ Rm for i = 1, . . . , N . Symmetric congestion games, in contrast to symmetric network congestion games, have been shown to be PLS-complete [8]. Hence, a natural question is: 1,4,6
a
d
2,3,6
4,6,7
b
e
2,3,5
1,2,7
c
f
1,2,8
Figure 1: Congestion game with 3 players. Each player chooses a vertex cover of the graph. 2
what polyhedral property captures this drop in complexity? If we make the additional assumption that the matrix A is totally unimodular (TU), i.e., each square submatrix of A has determinant equal to 0, +1, or −1, for each player i and fixed x−i , we can solve (1) in time polynomial in n, m [22]. However, finding a pure Nash equilibrium could still be PLS-complete. Our results. Our main contribution is to give an algorithm that runs in time polynomial in n, m and N , to find a pure Nash equilibrium of symmetric congestion games defined using a TU matrix A ∈ Rm×n . This algorithm subsumes, as a special case, the strongly polynomial-time algorithm of Fabrikant et al. for symmetric network congestion games. Our algorithm is structured into two phases. In the first phase, we solve an aggregated problem, where we minimize the potential function over the aggregated strategies of the players, to determine how many players should use each resource. In the second phase, we apply an algorithm that decomposes the optimal aggregated solution into the single players’ strategies. The properties of congestion games are crucial in the first phase: the existence of a potential function yields a reformulation of the equilibrium problem as a single optimization problem, and the cost structure of the game allows us to ignore the identities of the players and to only look the aggregated strategies. Total unimodularity is exploited in both phases: in the first phase it allows us to solve the aggregated problem with linear programming, and in the second phase it implies the so-called integer decomposition property[2], which is at the core of the decomposition algorithm. Concerning asymmetric congestion games, even if all the P i s share the same TU constraint matrix A and differ only in the right-hand side vectors bi , i = 1, . . . , N , we can directly show PLS-completeness, since asymmetric network congestion games can be cast in this setting. For a bipartite graph G = (V, E) we introduce four congestion games, where the strategies of each player are defined on a subgraph Gi of G as: (i) all matchings of Gi ; (ii) all edge covers of Gi ; (iii) all vertex covers of Gi ; (iv) all stable sets of Gi . In the symmetric case, when Gi = G for i = 1, . . . , N , the algorithm runs in time polynomial in |V |, |E| and N and has a combinatorial interpretations. In the asymmetric case, we show that all four games are PLS-complete through a reduction from POS NAE 3SAT. Organization of the paper. In Section 2 we present the classical notions of potential games, congestion games, polynomial local search problems, and we formally introduce totally unimodular congestion games and some of their combinatorial variants. In Section 3 we give a strongly polynomial-time algorithm for finding a pure Nash equilibrium in symmetric totally unimodular congestion games. We also discuss how this algorithm specializes when applied to several combinatorial games defined on graphs. In Section 4 we study other combinatorial games that can be reduced to totally unimodular congestion games and finally, in Section 5, we show that it is PLS-complete to find a pure Nash equilibrium in the asymmetric versions of all combinatorial congestion games previously introduced.
2
Preliminaries
Let (X, f ) be a game with N players, where X i is the strategy set of player i, X = X 1 × · · · × X N , f i : X → R is the payoff function of player i and f = (f 1 , . . . , f N ). We assume that, for all i ∈ {1, . . . , N }, X i is a finite set, and we call each element x ∈ X a state of the game. A pure Nash equilibrium is a state x = (x1 , . . . , xN ) such that for each i, f i (x1 , . . . , xi , . . . , xN ) ≥ f i (x1 , . . . , x ¯i , . . . , xN ) for any x ¯i ∈ X i . Congestion games. In a congestion game there is a finite set of resources R and, for each i = 1, . . . , N , X i ⊆ 2R . From now on we assume |R| = n and we identify each xi ∈ X i with its incidence vector in {0, 1}n . A nondecreasing delay function dj : {1, . . . , N } → Z is associated with each resource j ∈ {1, . . . , n}. For a resource j ∈ {1, . . . , n} and a state x ∈ X, denote by tj (x) the number of players using j in state x. The cost −f i (x) incurred by player i in state x ∈ X (i.e. the 3
negative of her payoff) is computed as the sum of the delays over all the resources selected by i, where the delay of resource j is dj (tj (x)). Note that, by definition, all the f i ’s are invariant under permutations of players. A congestion game is symmetric if all X i ’s are the same, i.e. all players have the same strategy set. In the following, we will also consider congestion games where a nonincreasing profit function pj : {1, . . . , N } → Z is associated with each resource. For each player i and state x ∈ X, the payoff of f i (x) is computed as the sum of the edge profits over all the resources selected by i, where the profit of resource j is pj (tj (x)). It can be easily checked that setting dj = −pj for all j = 1, . . . , n, we recover the standard definition of congestion games. In a classical paper [15], Rosenthal proved that any congestion game has a pure Nash equilibrium. The proof is based on the fact that congestion games are a subclass of (exact) potential games, and they admit the following potential function: φ(x) =
j (x) n tX X
dj (i).
(2)
j=1 i=1
Since φ is an exact potential function, for any x = (xi , x−i ) ∈ X, x ¯ = (¯ xi , x−i ) ∈ X, we have that φ(x)−φ(¯ x) = f i (¯ x)−f i (x). Thus a pure Nash equilibrium is a state x ∈ X such that, for each player i and x ¯i ∈ X i , φ(xi , x−i ) ≤ φ(¯ xi , x−i ). As a consequence, a global optimum of min{φ(x) : x ∈ X} is a pure Nash equilibrium of the congestion game. Totally unimodular congestion games. In totally unimodular (TU) congestion games, the sets X i correspond to the vectors xi that satisfy: Ai x i Q bi i
(3)
n
x ∈ {0, 1} , where bi ∈ Zm , and Ai ∈ {0, ±1}n×m is a given TU matrix, as defined above and treated in detail in [19]. The symbol Q indicates that each constraint of the system contains one of the three relations ≤, =, or ≥. A TU congestion game is symmetric if Ai = A and bi = b for every player i = 1, . . . , N . TU congestion games contain a wide variety of combinatorial congestion games, since for example incidence matrices of directed graphs, and of bipartite graphs are TU. Next, we define five combinatorial congestion games that we will consider in the remainder of the paper. • Network congestion games (N ). We are given a digraph D = (V, E), two nodes ri , si ∈ V for each player i, and a delay function with the arcs playing the role of the resources. The strategy set of player i is the set of all directed paths in D from ri to si . • Matching congestion games (M) (resp. edge cover congestion games (EC)). We are given a graph G = (V, E), a subgraph Gi = (V i , E i ) of G for each player i, and a nonincreasing profit (resp. nondecreasing delay) function with the edges playing the role of the resources. The strategy set X i of player i is the set of all matchings (resp. edge covers) in Gi . • Stable set congestion games (SS) (resp. vertex cover congestion games (VC)). We are given a graph G = (V, E), a subgraph Gi = (V i , E i ) of G for each player i, and a nonincreasing profit (resp. nondecreasing delay) function with the nodes playing the role of the resources. The strategy set X i of player i is the set of all stable sets (resp. vertex covers) in Gi . See [20] for more details on the above combinatorial problems. The complexity class PLS. A polynomial-time local search (PLS) problem [10] Π is a minimization1 problem defined by: (i) a set of instances IΠ , which are a polynomial-time recognizable subset 1 We
assume minimization without loss of generality.
4
of {0, 1}∗ ; (ii) for each instance I ∈ IΠ , a finite set of solutions SI , whose binary encoding is polynomial in the encoding of I; (iii) for each solution x ∈ SI , an integer c(x, I), the cost of x, and a subset N (x, I) ⊆ SI called the neighborhood of x. Moreover, three polynomial-time algorithms Q1 , Q2 and Q3 must be available: Q1 , given I ∈ IΠ , produces a solution Q1 (I) ∈ SI ; Q2 , given an instance I ∈ IΠ and a binary vector x, determines whether x ∈ SI and, if so, computes c(x, I); Q3 , given an instance I ∈ IΠ and a solution x ∈ SI , either outputs y ∈ N (x, I) with c(y, I) < c(x, I) or states that there is no such y ∈ N (x, I), implying that x is locally optimal. A problem Π is PLS-reducible to a problem Π0 if there are two polynomial-time computable functions h : IΠ → IΠ0 and g : Sh(I) → SI such that: h maps any instance I of Π to an instance h(I) of Π0 ; g maps any solution y of h(I) to a solution g(y) of I and for all I ∈ IΠ , if y ∗ is a local optimum for h(I), then g(y ∗ ) is a local optimum for I. A problem Π in PLS is PLS-complete if every problem in PLS is PLS-reducible to Π. A well-known PLS-complete problem is POS NAE 3SAT [18], i.e. not-all-equal-3SAT with positive literals only: an instance consists of clauses in conjunctive normal form; each clause has at most three positive literals and is assigned a positive cost; each clause is satisfied if its constituents do not all have the same value. A solution is a truth assignment, i.e. a 0/1 assignment to all variables; the cost of a solution, to be minimized, is the sum of the costs of the satisfied clauses; the neighborhood of a solution contains all solutions obtained by flipping the value of one variable. The local search problem is to find a truth assignment whose cost cannot be decreased by flipping a variable.
3
Polynomial solvability of the symmetric case
In this section we investigate the computational complexity of symmetric TU congestion games. Theorem 1. There is a strongly polynomial-time algorithm for finding a pure Nash equilibrium in symmetric TU congestion games. Proof. Since TU matrices are closed under adding copies of rows and multiplying rows by −1, we can assume without loss of generality that the strategy set of every player i is given by: X i = {xi : Axi ≥ b, xi ∈ {0, 1}n }. The algorithm computes the global optimum of min{φ(x) : x ∈ X},
(4)
where φ is the potential function of the congestion game and X = X 1 × · · · × X N is the strategy space. Since a global optimum of (4) is also a local optimum of the local search problem, the resulting state is a pure Nash equilibrium. In the first phase of our algorithm, we set up an aggregated problem as follows. Since the value of the potential function (2) only depends on how many players use a resource, we sum up the constraints corresponding to a given row P of matrix A for all players, and we define the aggregated N variables z ∈ [0, N ]n ∩ Zn such that zj = i=1 xij for each resource j. zj n X X min dj (i) : Az ≥ N b, z ∈ [0, N ]n ∩ Zn . (5) j=1 i=1
In the remainder of the proof, we find an optimum z¯ of (5), and then we show how to decompose it into a state x ¯ ∈ X such that z¯ = x ¯1 + · · · + x ¯N , with x ¯i ∈ X i . Since, for each state x ∈ X, the 1 N corresponding z = x + · · · + x is feasible for (5) and has the same objective value, we have that x ¯ is an optimum of (4). Next, to model the objective function of (5) as a linear function, we introduce variables y i ∈ {0, 1}n for i = 1, . . . , N , where yji = 1 if at least i players use resource j, and yji = 0 otherwise. Since 5
z = y 1 + · · · + y N for each resource j, we can write our aggregated problem as: minimize
n X N X
dj (i)yji
(6)
j=1 i=1
subject to
N X
Ay i ≥ N b
i=1
0 ≤ yi ≤ 1 i
i = 1, . . . , N
n
y ∈ {0, 1}
i = 1, . . . , N.
First, for each z feasible for (5), we define yji = 1 if i ≤ zj and yji = 0 otherwise. Note that y is feasible for (6) and with the same objective value as z. Moreover, for each y feasible for (6), the vector z = y 1 + · · · + y N is feasible for (5) and has objective value not larger than that of y. Therefore, the optimal solution y¯ of (6) yields an optimal solution z¯ of (5). We now show how to solve problem (6). First, the constraint matrix of the aggregated problem (6) is also TU, since it is of the form ( A |
A ··· {z
N times
A ). }
As the right hand side of the system is integral, the linear relaxation of the feasible set of the aggregated problem has only integral vertices (see for example Theorem 19.3 in [19]). Thus we can find an optimal solution y¯ of the aggregated problem via linear programming. Using Tardos’s [22] algorithm, this can be done in time polynomial in size(A) and in N , thus in time polynomial in n, m, N because all entries of A are in {0, ±1}. The vector z¯ = y¯1 + . . . y¯N is then an optimum of (5). In the remainder of the proof, we show how to derive from z¯ a state x ¯ with the same objective value in (4). Since A¯ z ≥ N b, and 0 ≤ z¯j ≤ N for every j = 1, . . . , n, z¯ is an integral vector in N P , where P = {x : 0 ≤ x ≤ 1, Ax ≥ b}. Since A is TU, the polyhedron P has the integer decomposition property, thus there exist integer vectors x ¯1 , . . . , x ¯N in P such that z¯ = x ¯1 + · · · + x ¯N . Following Baum and Trotter [2], we show how to obtain such vectors in strongly polynomial time. We show how to find an integer vector x ¯1 in P such that z¯ − x ¯1 is an integer vector in (N − 1)P . In order to do so, we define P 1 = {s : 0 ≤ s ≤ 1, z¯ − (N − 1) ≤ s ≤ z¯, b ≤ As ≤ A¯ z − (N − 1)b}.
(7)
The polyhedron P 1 is nonempty since it contains z¯/N , and is integral since its constraint matrix is TU. Again using Tardos’s [22] algorithm, we can find a vertex x ¯1 of P 1 in time polynomial in n, m. By applying the above argument recursively N times, we obtain integral vectors x ¯1 , x ¯2 , . . . , x ¯N in P with z¯ = x ¯1 + · · · + x ¯N . Therefore, x ¯ is a pure Nash equilibrium. The total running time of the algorithm is polynomial in n, m, N .
3.1
Consequences for combinatorial problems
In this section we show that the combinatorial game N , and games M, EC, SS, VC on bipartite graphs are TU congestion games. We then explain, from a combinatorial point of view, when such games are symmetric. In such cases, Theorem 1 implies that one can find in strongly polynomial time a pure Nash equilibrium. Proposition 1. Congestion games N , and congestion games M, EC, SS, VC on bipartite graphs are TU congestion games.
6
Proof. Network congestion games are TU congestion games, since xi is the incidence vector of a dipath from ri to si in digraph D = (V, E) if and only if Axi = bi , xi ∈ {0, 1}E , where A is the V × E incidence matrix of D, and bi has entry −1 corresponding to node ri , entry +1 corresponding to node si , and all other entries are 0. M and EC on bipartite graphs are TU congestion games, since xi is the incidence vector of a i matching (resp. edge cover) in Gi = (V i , E i ) if and only if Ai xi ≤ 1 (resp. Ai xi ≥ 1), xi ∈ {0, 1}E , where Ai is the V i × E i incidence matrix of Gi . SS and VC on bipartite graphs are TU congestion games, since xi is the incidence vector of a stable set (resp. vertex cover) in Gi = (V i , E i ) if and only if Ai xi ≤ 1 (resp. Axi ≥ 1), xi ∈ {0, 1}Vi , where Ai is the E i × V i incidence matrix of Gi . Recall that a TU congestion game is symmetric if Ai = A and bi = b for every player i = 1, . . . , N . As a consequence, games N are symmetric if all players have the same origin ri = r and destination si = s. Games M, EC, SS, VC are symmetric if all players act on the same subgraph, i.e. Gi = G for all i = 1, . . . , N . Theorem 1 then directly implies the following corollary: Corollary 1. There is a strongly polynomial-time algorithm for finding a pure Nash equilibrium in the symmetric case of games N , and of games M, EC, SS, VC on bipartite graphs. Interestingly, the algorithm given in the proof of Theorem 1 has some nice combinatorial interpretations in the above combinatorial games. For the symmetric case of N , we recover the algorithm described in [8]. Given a digraph D = (V, E) and arc delay functions de , we construct a new digraph D0 = (V, E 0 ) by replacing each arc e ∈ E with N parallel arcs e1 , . . . , eN between the same nodes, with delays de (1), . . . , de (N ) and capacity 1. Solving problem (6) is equivalent to finding an integer flow F of value N from origin r to destination s of minimum cost in the digraph D0 . This is a ˜ be the subgraph of D obtained by minimum cost flow problem (see Ch. 12 in [20]). Now let D ¯ be the set deleting arc e ∈ E if no edge among e1 , . . . , eN is used by the flow F . Moreover, let E of arcs e ∈ E such that all arcs e1 , . . . , eN are used by the flow F . Finding an integer vector in (7) ˜ that contains all arcs in is then equivalent to finding a directed path from r to s in the digraph D ¯ In the case where there are no directed cycles of negative delay in D0 , this is equivalent to just E. ˜ finding a directed path from r to s in D. For the remaining combinatorial congestion games, we will need the following construction. Given bipartite graph G = (V, E) and profits pe (or delays de ), we construct a new bipartite graph G0 = (V, E 0 ) by replacing each edge e ∈ E with N parallel edges e1 , . . . , eN between the same nodes, with profits pe (1), . . . , pe (N ) (or delays de (1), . . . , de (N )). For games M, solving problem (6) is equivalent to finding a maximum profit subset F of E 0 such that each node v in V is incident to at most N edges in F . This is a simple b-matching problem (see ˜ be the subgraph of G obtained by deleting edge e ∈ E if no edge among Ch. 21 in [20]). Now let G 1 N e , . . . , e is in F . Moreover, let U be the set of nodes that have degree N in (V, F ). Finding an ˜ covering all nodes in U . integer vector in (7) is then equivalent to finding a matching in the graph G For games EC, solving problem (6) is equivalent to finding a minimum cost subset F of E 0 such that each node v in V is incident to at least N edges in F . This is a simple b-edge cover problem ˜ be the subgraph of G obtained by deleting edge e ∈ E if either no (see Ch. 21 in [20]). Now let G 1 N edge among e , . . . , e is in F , or if if all edges e1 , . . . , eN are in F . Finding an integer vector in (7) ˜ such that for every node v ∈ V , v is is then equivalent to finding a set of edges C in in the graph G incident to a number of edges in C between bounds lv and uv that can be easily calculated. For games VC and SS, one can find an integer vector x ¯1 ∈ P 1 by finding a perfect vertex cover, ˜ of G. For vertex cover congestion i.e. a vertex cover that is also a stable set, in a suitable subgraph G ˜ games, G is the subgraph of G obtained by first deleting nodes v ∈ V if z¯v = 0 (these nodes will have x ¯1v = 0), then by deleting all edges uv with z¯u + z¯v 6= N , and finally by removing the isolated nodes ˜ is the subgraph of G obtained (these nodes will have x ¯1v = 1). For stable set congestion games, G 7
by first deleting nodes v ∈ V if z¯v = N (these nodes will have x ¯1v = 1), then by deleting all edges uv with z¯u + z¯v 6= N , and finally by removing the isolated nodes (these nodes will have x ¯1v = 0).
4
More combinatorial games
In this section we consider some variants of the combinatorial games previously defined that do not appear to be TU congestion games, but where we can still find a pure Nash equilibrium in strongly polynomial time. A maximum cardinality matching congestion games (C − M) is a variant of M where each set X i consists only of the maximum cardinality matchings in Gi . Similarly, in minimum cardinality edge cover congestion games (C − EC) each set X i consists of the minimum cardinality edge covers in Gi ; In maximum cardinality stable set congestion games (C − SS) each set X i consists of the maximum cardinality stable sets in Gi ; In minimum cardinality vertex cover congestion games (C − VC) each set X i consists of the minimum cardinality vertex covers in Gi . Proposition 2. Given an instance I of a C − M (resp. C − EC, C − SS, C − VC) on a graph G, one can construct in strongly polynomial time an instance h(I) of a M (resp. EC, SS, VC) on the same graph G, so that a pure Nash equilibrium in h(I) is also a pure Nash equilibrium in I. Proof. We give the reduction for matching games, the other ones being analogous. To map an instance I of C − M to an instance h(I) of M, we only need to modify the edge profits. Let G = (V, E). Let pe (i), i = 1, . . . , N , be the given profits in I, and let ∆ = max{|pe (i)| : e ∈ E, i = 1, . . . , N }. We set the profits in h(I) to be p0e (i) = pe (i) + 2|E|∆ + 1. It can be checked that for every player i, the payoff in h(I) corresponding to a matching in Gi that is of maximum cardinality will always be strictly larger than a payoff in h(I) corresponding to a matching in Gi that is not of maximum cardinality. We claim that a pure Nash equilibrium x for h(I) is also a pure Nash equilibrium for I. By construction, the strategy of each player i in x corresponds to a maximum cardinality matching M i in Gi . The payoff f˜i (x) of player i in h(I) is f˜i (x) = f i (x) + k(2|E|∆ + 1), where f i (x) is the payoff of player i in I, and k is the cardinality of a maximum cardinality matching in Gi . By contradiction, assume that there is another maximum cardinality matching N i in Gi such that in the state x0 obtained from x by swapping M i with N i , player i has better payoff in I, i.e., f i (x0 ) > f i (x). This implies f˜i (x0 ) = f i (x0 ) + k(2|E|∆ + 1) > f i (x) + k(2|E|∆ + 1) = f˜i (x), contradicting the fact that x is a pure Nash equilibrium for h(I). The given reduction is strongly polynomial. By Proposition 2 and Corollary 1 we obtain the following: Corollary 2. There is a strongly polynomial-time algorithm for finding a pure Nash equilibrium in the symmetric case of the following games on bipartite graphs: C − M, C − EC, C − SS, C − VC.
5
PLS-completeness of the asymmetric case
In this section we focus on asymmetric TU congestion games, i.e. congestion games in their most general form (3). The next Proposition follows directly from PLS-completeness of asymmetric network congestion games [8], see proof of Proposition 1. Proposition 3. Asymmetric TU congestion games in the form (3) are PLS-complete, even if A = Ai , i = 1, . . . , N . We remark that, in fact, all the asymmetric variants of problems M, EC, SS and VC on bipartite graphs can be written as TU congestion games with A = Ai , for i = 1, . . . , N . Our goal is to prove that all these asymmetric TU congestion are PLS-complete. 8
To this purpose we define the perfect matching congestion game (PM) as a variant of M where for every player i, the subgraph Gi admits a perfect matching, and where the set X i consists only of the perfect matchings in Gi . PM is equivalent to (i) C − M where all subgraphs Gi admit a perfect matching; (ii) C − EC where all subgraphs Gi admit a perfect matching, and the edge delays are the negative of the profits. Similarly, the perfect vertex cover congestion game (PVC) is a variant of VC where for every player i, the subgraph Gi admits a perfect vertex cover, i.e., a vertex cover that is also a stable set, and where the set X i consists only of the perfect vertex covers in Gi . PVC is equivalent to (i) C − VC where all subgraphs Gi admit a perfect vertex cover; (ii) C − SS where all subgraphs Gi admit a perfect matching and the vertex profits are the negative of the delays. We first show that the asymmetric PM on a bipartite graph is PLS-complete. Theorem 2. It is PLS-complete to find a pure Nash equilibrium in the asymmetric perfect matching congestion game on a bipartite graph. Proof. We give a PLS-reduction of POS NAE 3SAT to an asymmetric PM on a bipartite graph G. First, we define a map h from any instance of POS NAE 3SAT to an instance of an asymmetric PM on a bipartite graph G. Given an instance I of POS NAE 3SAT, we construct a congestion game h(I) as follows. Denote by C = {c1 , . . . , cn } the clauses of I and by {x1 , . . . , xN } its variables. Let wj be the cost of clause cj and, for a truth assignment x of I denote by w(x) the cost of x. Each variable of POS NAE 3SAT is a player of the PM and each NAE clause is a set of resources, i.e. a set of edges. Precisely, for each NAE clause cj we build the graph “gadget” in Fig. 2a. The graph gadget of clause cj is a 4-cycle uj , vj , zj , v¯j , uj . vj+1
vj
vj +
2
vj zj
zj
uj
zj+1
uj+1
uj +
2
zj +
uj
2
v¯j
v¯j
v¯j+1
v¯j +
2
(a)
(b)
Figure 2: Reduction from POS NAE 3SAT to asymmetric PM on a bipartite graph. Let mj ∈ {1, 2, 3} denote the number of variables in cj . The edge profits are defined as follows: • If e = v¯j uj or e = v¯j zj , then pe (i) = 0 for i = 1, . . . , N ; • If e = uj vj and cj contains at least a constant equal to 1, then pe (i) = 0 for i = 1, . . . , N ; otherwise, pe (i) = 0 for i = 1, . . . , mj − 1 and pe (i) = −wj for i = mj , . . . , N ; • If e = vj zj and cj contains at least a constant equal to 0, then pe (i) = 0 for i = 1, . . . , N ; otherwise, pe (i) = 0 for i = 1, . . . , mj − 1 and pe (i) = −wj for i = mj , . . . , N . Now, we build graph G as follows: for any two clauses cj and cj+1 , j = 1, . . . , n−1 we identify zj and uj+1 ; moreover, we identify zn and u1 , see Fig. 2b. Clearly, G is a bipartite graph with bipartitions {vj , v¯j }j=1,...,n and {uj }j=1,...,n . For each i = 1, . . . , N , let C(i) denote the set of clauses containing variable xi and let Vi = {uj , zj , j = 1, . . . , n} ∪ {vj : cj ∈ C(i)} ∪ {¯ vj : cj ∈ / C(i)}. We assign to
9
player i the subgraph Gi of G induced by nodes in Vi . This shows how to map I to an instance h(I) of asymmetric PM on a bipartite graph. Next, we define a map g from states of h(I) to truth assignments of I. A state of h(I) is a set of N perfect matchings on graphs Gi , i = 1, . . . , N . Note that each subgraph Gi is a cycle of length 2n that admits two perfect matchings: M0i = {uj vj : cj ∈ C(i)} ∪ {uj v¯j : cj ∈ / C(i)} and vj zj : cj ∈ / C(i)}. Let gi : {M0i , M1i } → {0, 1} such that gi (M0i ) = 0 and M1i = {vj zj : cj ∈ C(i)} ∪ {¯ gi (M1i ) = 1. We map strategy M i ∈ {M0i , M1i } of player i to xi = gi (M i ). Setting g = (g1 , . . . , gN ) shows that any state of h(I) is mapped to a truth assignment of I, and that the mapping is bijective. Finally, we need to show that any pure Nash equilibrium of h(I) maps to a local minimum of I. First, we remark that, for any truth assignment x of I, each x0 ∈ N (x, I) obtained by flipping variable xi , i ∈ {1, . . . , N } is in one-to-one correspondence with the state obtained from g −1 (x) after the defection of player i. Now, let M 1 , . . . , M N be a a pure Nash equilibrium of h(I), where M i ∈ {M0i , M1i } for all i = 1, . . . , N , and denote by y the corresponding state. Then, for any state y 0 obtained from y by switching perfect matching M i , we have that fi (y) − fi (y 0 ) ≥ 0, where fi denotes the payoff of player i in PM. By construction, for x = g(y) and x0 = g(y 0 ) we have that w(x0 ) − w(x) = fi (y) − fi (y 0 ) ≥ 0. This proves that x is a local optimum of I. Theorem 2 directly implies the following result. Corollary 3. It is PLS-complete to find a pure Nash equilibrium (i) in the asymmetric matching congestion game on a bipartite graph; (ii) in the asymmetric edge cover congestion game on a bipartite graph. Proof. (i) Since PM is equivalent to C − M when all subgraphs Gi admit a perfect matching, we have that asymmetric C − M is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 2. (ii) Since PM is equivalent to C − EC when all subgraphs Gi admit a perfect matching, we have that asymmetric C − EC is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 2. We now show that the asymmetric PVC on a bipartite graph is PLS-complete. Theorem 3. It is PLS-complete to find a pure Nash equilibrium in the asymmetric perfect vertex cover congestion game on a bipartite graph. Proof. We give a PLS-reduction of POS NAE 3SAT to an asymmetric PVC on a bipartite graph G. The proof structure is similar to that of Theorem 3, thus here we only outline the main differences. Again, the variables of POS NAE 3SAT map to players of PVC and clauses map to a set of resources. For each NAE clause cj we build the graph “gadget” in Fig. 3a, that is a 8-cycle uj , sj , vj , tj , zj , t¯j , v¯j , s¯j , uj . For mj ∈ {1, 2, 3} the vertex profits are defined as: • If u ∈ / {sj , vj }, then du (i) = 0 for i = 1, . . . , N ; • If u = sj and cj contains at least a constant equal to 1, then du (i) = 0 for i = 1, . . . , N ; otherwise, du (i) = 0 for i = 1, . . . , mj − 1 and du (i) = wj for i = mj , . . . , N ; • If u = vj and cj contains at least a constant equal to 0, then du (i) = 0 for i = 1, . . . , N ; otherwise, du (i) = 0 for i = 1, . . . , mj − 1 and du (i) = wj for i = mj , . . . , N . We build a bipartite graph G by identifying zn and u1 and zj and uj+1 for j = 1, . . . , n − 1, see Fig. 3b. For each i = 1, . . . , N , we let Vi = {uj , zj , j = 1, . . . , n} ∪ {sj , vj , tj : cj ∈ C(i)} ∪ {¯ sj , v¯j , t¯j : cj ∈ / C(i)} and we define Gi as the subgraph of G induced by nodes in Vi . Since each Gi is a cycle of length 4n, it admits exactly two perfect vertex covers: W0i = {sj , tj : cj ∈ C(i)} ∪ {¯ sj , t¯j : cj ∈ / C(i)} and W1i = {uj , vj : cj ∈ C(i)} ∪ {uj , v¯j : cj ∈ / C(i)}. We define i i gi : {W0 , W1 } → {0, 1} such that gi (W0i ) = 0 and gi (W1i ) = 1 and we map strategy W i ∈ {W0i , W1i } of player i to xi = gi (W i ). Let y be a Nash equilibrium of PVC corresponding to vertex covers W 1 , . . . , W N , and denote by −fi the cost function of player i. Then, for any state y 0 obtained by switching perfect vertex cover 10
vj+1
vj
sj+1
tj
vj
uj
2
zj
zj t¯j
s¯j v¯j
2
tj +
sj
tj
sj
vj +
tj+1 sj +2 zj+1
uj+1
uj ¯tj
s¯j v¯j
uj +
2
zj +
s¯j +
s¯j+1
t¯j+1 v¯j+1
2
2
t¯j +
2
v¯j +
2
(a)
(b)
Figure 3: Reduction from POS NAE 3SAT to asymmetric PVC on a bipartite graph. W i , we have that fi (y 0 ) − fi (y) ≥ 0. By construction, for x = g(y) and x0 = g(y 0 ) it follows that w(x0 ) − w(x) = fi (y 0 ) − fi (y) ≥ 0. Theorem 3 directly implies the following result. Corollary 4. It is PLS-complete to find a pure Nash equilibrium (i) in the asymmetric vertex cover congestion game on a bipartite graph; (ii) in the asymmetric stable set congestion game on a bipartite graph. Proof. (i) Since PVC is equivalent to C − VC when all subgraphs Gi admit a perfect vertex cover, we have that asymmetric C − VC is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 3. (ii) Since PVC is equivalent to C − SS when all subgraphs Gi admit a perfect vertex cover, we have that asymmetric C − SS is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 3.
6
Conclusion
We have introduced the notion of TU congestion games and we have shown that finding a pure Nash equilibrium can be done in strongly polynomial time in the symmetric case. We have also demonstrated that, for several combinatorial variants of TU Congestion Games, the asymmetric case is PLS-complete. Our results suggest that the computational complexity of a game may depend on the complexity of a polyhedral representation of the strategy sets of players. A crucial property in the design of our algorithm is the integral decomposition property, allowing us to decompose an aggregated strategy that minimizes the potential function into a pure Nash equilibrium. This property may also hold if the constraint matrix of the symmetric TU congestion game is not TU. Open questions involve extensions of our results to non-TU congestion games and approximation results in the general case.
References ´ [1] Elliot Anshelevich, Anirban Dasgupta, Jon Kleinberg, Eva Tardos, Tom Wexler, and Tim Roughgarden. The price of stability for network design with fair cost allocation. In Proceedings
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of the 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’04, pages 295–304, Washington, DC, USA, 2004. IEEE Computer Society. [2] Steven P. Baum and Leslie E. Trotter Jr. Integer rounding and polyhedral decomposition for totally unimodular systems. In Rudolf Henn, Bernhard Korte, and Werner Oettli, editors, Optimization and Operations Research, volume 157 of Lecture Notes in Economics and Mathematical Systems, pages 15–23. Springer Berlin Heidelberg, 1978. [3] Anand Bhalgat, Tanmoy Chakraborty, and Sanjeev Khanna. Approximating pure Nash equilibrium in cut, party affiliation, and satisfiability games. In Proceedings of the 11th ACM Conference on Electronic Commerce, EC ’10, pages 73–82, New York, NY, USA, 2010. ACM. [4] Ioannis Caragiannis, Angelo Fanelli, Nick Gravin, and Alexander Skopalik. Efficient computation of approximate pure Nash equilibria in congestion games. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS ’11, pages 532– 541, Washington, DC, USA, 2011. IEEE Computer Society. [5] Ioannis Caragiannis, Angelo Fanelli, Nick Gravin, and Alexander Skopalik. Approximate pure Nash equilibria in weighted congestion games: Existence, efficient computation, and structure. In Proceedings of the 13th ACM Conference on Electronic Commerce, EC ’12, pages 284–301, New York, NY, USA, 2012. ACM. [6] Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player Nash equilibria. J. ACM, 56(3):14:1–14:57, May 2009. [7] Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195–259, 2009. [8] Alex Fabrikant, Christos H. Papadimitriou, and Kunal Talwar. The complexity of pure Nash equilibria. In Proceedings of STOC ’04, 2004. [9] Charles A. Holt and Alvin E. Roth. The Nash equilibrium: A perspective. Proceedings of the National Academy of Sciences, 101(12):3999–4002, 2004. [10] David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? Journal of Computer and System Sciences, 37(1):79 – 100, 1988. [11] Dov Monderer and Lloyd S. Shapley. Potential games. Games and Economic Behavior, 14(1):124 – 143, 1996. [12] John Nash. Equilibrium points in n-person games. In Proceedings of National Academy of Sciences, volume 36, pages 48–49, 1950. [13] John Nash. Non-cooperative games. Annals of Mathematics, 54(2):286–295, 1951. ´ Tardos. Approximate pure Nash equilibria via Lov´asz local lemma. In [14] Th` anh Nguyen and Eva Stefano Leonardi, editor, Internet and Network Economics, volume 5929 of Lecture Notes in Computer Science, pages 160–171. Springer Berlin Heidelberg, 2009. [15] Robert W. Rosenthal. A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory, 2:65–67, 1973. [16] Robert W. Rosenthal. The network equilibrium problem in integers. Networks, 3(1):53–59, 1973. ´ Tardos. How bad is selfish routing? J. ACM, 49(2):236–259, March [17] Tim Roughgarden and Eva 2002. 12
[18] Alejandro A. Sch¨ affer and Mihalis Yannakakis. Simple local search problems that are hard to solve. SIAM J. Comput., 20(1):56–87, February 1991. [19] Alexander Schrijver. Theory of Linear and Integer Programming. Wiley, Chichester, 1986. [20] Alexander Schrijver. Combinatorial Optimization. Polyhedra and Efficiency. Springer-Verlag, Berlin, 2003. [21] Alexander Skopalik and Berthold V¨ocking. Inapproximability of pure Nash equilibria. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC ’08, pages 355–364, New York, NY, USA, 2008. ACM. ´ Tardos. A strongly polynomial algorithm to solve combinatorial linear programs. Operations [22] Eva Research, 34(2):250–256, 1986. [23] Adrian Vetta. Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In Foundations of Computer Science, 2002. Proceedings. The 43rd Annual IEEE Symposium on Foundations of Computer Science, pages 416–425, 2002.
13
Michael Ferris†
Carla Michini‡
arXiv:1511.02784v1 [cs.GT] 9 Nov 2015
November 10, 2015
Abstract We investigate a new class of congestion games, called Totally Unimodular Congestion Games, in which the strategies of each player are expressed as binary vectors lying in a polyhedron defined using a totally unimodular constraint matrix and an integer right-hand side. We study both the symmetric and the asymmetric variants of the game. In the symmetric variant, all players have the same strategy set, i.e. the same constraint matrix and right-hand side. Network congestion games are an example of this class. Fabrikant et al. proved that a pure Nash equilibrium of symmetric network congestion games can be found in strongly polynomial time, while the asymmetric network congestion games are PLS-complete. We give a strongly polynomial-time algorithm to find a pure Nash equilibrium of any symmetric totally unimodular congestion game. We also identify four totally unimodular congestion games, where the players’ strategy sets are matchings, vertex covers, edge covers and stable sets of a given bipartite graph. For these games we derive specialized combinatorial algorithms to find a pure Nash equilibrium in the symmetric variant, and show the asymmetric variant is PLS-complete.
1
Introduction
A central problem of Algorithmic Game Theory concerns the existence of Nash equilibria and the design of polynomial-time algorithms for their computation. Nash equilibria are a very powerful and well-studied solution concept, and they have had tremendous impact in economics and social sciences [9]. A mixed strategy is a probability distribution over the pure strategies of a player. In his celebrated work [12, 13], John Nash proved the existence of mixed equilibria for any game with a finite set of strategies. More recently it has been shown that finding a mixed equilibrium is PPAD-complete [6, 7]. In some applications mixed Nash equilibria seems to have no natural interpretation, and thus, pure Nash equilibria are sometimes a more suitable solution concept than mixed Nash equilibria [23, 1]. Unfortunately, pure Nash equilibria are not guaranteed to exist, and even if they do, they are often hard to compute. Thus, much effort has been devoted to designing polynomial-time algorithms to determine an approximate pure Nash equilibrium [3, 4, 5, 14]. On the positive side, there are classes of games that are known to posses pure Nash equilibria. A prominent example are potential games, originally introduced by Monderer and Shapley [11]. The key property of potential games is the existence of a potential function, i.e. a function defined on the joint strategy sets of the players, and such that, if any player unilaterally deviates from her strategy, the change in her payoff is equal to the change in the potential function. A pure Nash equilibrium of a potential game can be found with a local search algorithm that minimizes the potential function over the strategy space of the game, where each iteration of the algorithm corresponds to an improving step of a player. Even if this algorithm is finite, it could take an exponential number of iterations ∗ Department
of Industrial and Systems Engineering, University of Wisconsin-Madison. Email: [email protected] of Computer Sciences, University of Wisconsin-Madison. Email: [email protected] ‡ Wisconsin Institute for Discovery, University of Wisconsin-Madison. Email: [email protected]
† Department
1
to reach a local optimum, or equivalently, a pure Nash equilibrium. Potential games belong to the complexity class Polynomial Local Search (PLS), introduced by Johnson, Papadimitriou and Yannakakis [10, 18]. In fact, this class includes all problems with a local search algorithm where each improving step can be performed in polynomial time. Many families of potential games have been shown to be PLS-complete, suggesting that a polynomial-time algorithm to find a pure Nash equilibrium is unlikely to exist for these games in general. In this work, we focus on congestion games, a class of potential games that has been widely investigated in the literature [15, 17, 21, 8, 16, 4, 5], and we provide new insight on their computational complexity using a polyhedral approach. In a congestion game, a set of resources is given, and each player selects a feasible subset of the resources in order to minimize her cost function. The cost of a player’s strategy is the sum of the delays of the resources selected by the player, and the delay of each resource is a function of the total number of players using it. Feasible means that not all possible combinations of resources belong to the strategy set of a player. In particular, we will assume that the strategies of each player are implicitly given through a polyhedral representation. For example, we can define a congestion game where the set of resources is the node set of a bipartite graph G = (V, E), and the players’ strategies are the vertex covers of G. Figure 1 shows an instance of this game, where there are 3 players and each node has different delays when it is used by 1,2 or 3 players. If player 1 chooses {a, e, f }, player 2 chooses {c, d, e}, and player 3 chooses {d, e, f }, then their costs are 8, 9 and 10, respectively. However, this is not a pure Nash equilibrium, since player 1 could unilaterally deviate and choose vertex cover {a, b, c}, decreasing her cost by 1. Another example are network congestion games, where the resources are the arcs of a given digraph D = (V, E) and the strategies of each player i are all (ri , si )-paths in D, ri , si ∈ V . Fabrikant et al. [8] gave an algorithm to find a pure Nash equilibrium in symmetric network congestion games, i.e. when all players share the same origin-destination pair. The algorithm is based on a reduction to minimum cost flow and runs in strongly polynomial time. Fabrikant et al. also showed that asymmetric network congestion games are PLS-complete. Polyhedral approach. In a general congestion game with N players and n resources, let X i = {χxi : xi is a strategy of player i}, where χ denotes the incidence vector, and let P i = {xi ∈ [0, 1]n : Ai xi ≥ bi } be a polyhedron such that P i ∩ {0, 1}n = X i . We can formulate the problem of player i as Pn −i i minxi j=1 cj (x )xj (1) i i s.t. x ∈ P ∩ {0, 1}n , where x−i is the vector encoding the strategies of all the other players and cj (x−i ) is the cost that i would incur by selecting resource j, given x−i . Note that the objective of (1) is linear for fixed x−i . Our goal is to understand what properties of P i affect the computational complexity of finding a pure Nash equilibrium. Clearly, if X i is explicitly given for all i, we can find a pure Nash equilibrium by exhaustive search of the payoff matrix. In polyhedral terms, this would imply that, for each i and fixed x−i , we could solve (1) using the vertex description of conv(X i ). The complexity of this linear programming problem would indeed be polynomial in the input size. We first consider the class of symmetric congestion games, i.e. congestion games with Ai = A ∈ Rm×n and bi = b ∈ Rm for i = 1, . . . , N . Symmetric congestion games, in contrast to symmetric network congestion games, have been shown to be PLS-complete [8]. Hence, a natural question is: 1,4,6
a
d
2,3,6
4,6,7
b
e
2,3,5
1,2,7
c
f
1,2,8
Figure 1: Congestion game with 3 players. Each player chooses a vertex cover of the graph. 2
what polyhedral property captures this drop in complexity? If we make the additional assumption that the matrix A is totally unimodular (TU), i.e., each square submatrix of A has determinant equal to 0, +1, or −1, for each player i and fixed x−i , we can solve (1) in time polynomial in n, m [22]. However, finding a pure Nash equilibrium could still be PLS-complete. Our results. Our main contribution is to give an algorithm that runs in time polynomial in n, m and N , to find a pure Nash equilibrium of symmetric congestion games defined using a TU matrix A ∈ Rm×n . This algorithm subsumes, as a special case, the strongly polynomial-time algorithm of Fabrikant et al. for symmetric network congestion games. Our algorithm is structured into two phases. In the first phase, we solve an aggregated problem, where we minimize the potential function over the aggregated strategies of the players, to determine how many players should use each resource. In the second phase, we apply an algorithm that decomposes the optimal aggregated solution into the single players’ strategies. The properties of congestion games are crucial in the first phase: the existence of a potential function yields a reformulation of the equilibrium problem as a single optimization problem, and the cost structure of the game allows us to ignore the identities of the players and to only look the aggregated strategies. Total unimodularity is exploited in both phases: in the first phase it allows us to solve the aggregated problem with linear programming, and in the second phase it implies the so-called integer decomposition property[2], which is at the core of the decomposition algorithm. Concerning asymmetric congestion games, even if all the P i s share the same TU constraint matrix A and differ only in the right-hand side vectors bi , i = 1, . . . , N , we can directly show PLS-completeness, since asymmetric network congestion games can be cast in this setting. For a bipartite graph G = (V, E) we introduce four congestion games, where the strategies of each player are defined on a subgraph Gi of G as: (i) all matchings of Gi ; (ii) all edge covers of Gi ; (iii) all vertex covers of Gi ; (iv) all stable sets of Gi . In the symmetric case, when Gi = G for i = 1, . . . , N , the algorithm runs in time polynomial in |V |, |E| and N and has a combinatorial interpretations. In the asymmetric case, we show that all four games are PLS-complete through a reduction from POS NAE 3SAT. Organization of the paper. In Section 2 we present the classical notions of potential games, congestion games, polynomial local search problems, and we formally introduce totally unimodular congestion games and some of their combinatorial variants. In Section 3 we give a strongly polynomial-time algorithm for finding a pure Nash equilibrium in symmetric totally unimodular congestion games. We also discuss how this algorithm specializes when applied to several combinatorial games defined on graphs. In Section 4 we study other combinatorial games that can be reduced to totally unimodular congestion games and finally, in Section 5, we show that it is PLS-complete to find a pure Nash equilibrium in the asymmetric versions of all combinatorial congestion games previously introduced.
2
Preliminaries
Let (X, f ) be a game with N players, where X i is the strategy set of player i, X = X 1 × · · · × X N , f i : X → R is the payoff function of player i and f = (f 1 , . . . , f N ). We assume that, for all i ∈ {1, . . . , N }, X i is a finite set, and we call each element x ∈ X a state of the game. A pure Nash equilibrium is a state x = (x1 , . . . , xN ) such that for each i, f i (x1 , . . . , xi , . . . , xN ) ≥ f i (x1 , . . . , x ¯i , . . . , xN ) for any x ¯i ∈ X i . Congestion games. In a congestion game there is a finite set of resources R and, for each i = 1, . . . , N , X i ⊆ 2R . From now on we assume |R| = n and we identify each xi ∈ X i with its incidence vector in {0, 1}n . A nondecreasing delay function dj : {1, . . . , N } → Z is associated with each resource j ∈ {1, . . . , n}. For a resource j ∈ {1, . . . , n} and a state x ∈ X, denote by tj (x) the number of players using j in state x. The cost −f i (x) incurred by player i in state x ∈ X (i.e. the 3
negative of her payoff) is computed as the sum of the delays over all the resources selected by i, where the delay of resource j is dj (tj (x)). Note that, by definition, all the f i ’s are invariant under permutations of players. A congestion game is symmetric if all X i ’s are the same, i.e. all players have the same strategy set. In the following, we will also consider congestion games where a nonincreasing profit function pj : {1, . . . , N } → Z is associated with each resource. For each player i and state x ∈ X, the payoff of f i (x) is computed as the sum of the edge profits over all the resources selected by i, where the profit of resource j is pj (tj (x)). It can be easily checked that setting dj = −pj for all j = 1, . . . , n, we recover the standard definition of congestion games. In a classical paper [15], Rosenthal proved that any congestion game has a pure Nash equilibrium. The proof is based on the fact that congestion games are a subclass of (exact) potential games, and they admit the following potential function: φ(x) =
j (x) n tX X
dj (i).
(2)
j=1 i=1
Since φ is an exact potential function, for any x = (xi , x−i ) ∈ X, x ¯ = (¯ xi , x−i ) ∈ X, we have that φ(x)−φ(¯ x) = f i (¯ x)−f i (x). Thus a pure Nash equilibrium is a state x ∈ X such that, for each player i and x ¯i ∈ X i , φ(xi , x−i ) ≤ φ(¯ xi , x−i ). As a consequence, a global optimum of min{φ(x) : x ∈ X} is a pure Nash equilibrium of the congestion game. Totally unimodular congestion games. In totally unimodular (TU) congestion games, the sets X i correspond to the vectors xi that satisfy: Ai x i Q bi i
(3)
n
x ∈ {0, 1} , where bi ∈ Zm , and Ai ∈ {0, ±1}n×m is a given TU matrix, as defined above and treated in detail in [19]. The symbol Q indicates that each constraint of the system contains one of the three relations ≤, =, or ≥. A TU congestion game is symmetric if Ai = A and bi = b for every player i = 1, . . . , N . TU congestion games contain a wide variety of combinatorial congestion games, since for example incidence matrices of directed graphs, and of bipartite graphs are TU. Next, we define five combinatorial congestion games that we will consider in the remainder of the paper. • Network congestion games (N ). We are given a digraph D = (V, E), two nodes ri , si ∈ V for each player i, and a delay function with the arcs playing the role of the resources. The strategy set of player i is the set of all directed paths in D from ri to si . • Matching congestion games (M) (resp. edge cover congestion games (EC)). We are given a graph G = (V, E), a subgraph Gi = (V i , E i ) of G for each player i, and a nonincreasing profit (resp. nondecreasing delay) function with the edges playing the role of the resources. The strategy set X i of player i is the set of all matchings (resp. edge covers) in Gi . • Stable set congestion games (SS) (resp. vertex cover congestion games (VC)). We are given a graph G = (V, E), a subgraph Gi = (V i , E i ) of G for each player i, and a nonincreasing profit (resp. nondecreasing delay) function with the nodes playing the role of the resources. The strategy set X i of player i is the set of all stable sets (resp. vertex covers) in Gi . See [20] for more details on the above combinatorial problems. The complexity class PLS. A polynomial-time local search (PLS) problem [10] Π is a minimization1 problem defined by: (i) a set of instances IΠ , which are a polynomial-time recognizable subset 1 We
assume minimization without loss of generality.
4
of {0, 1}∗ ; (ii) for each instance I ∈ IΠ , a finite set of solutions SI , whose binary encoding is polynomial in the encoding of I; (iii) for each solution x ∈ SI , an integer c(x, I), the cost of x, and a subset N (x, I) ⊆ SI called the neighborhood of x. Moreover, three polynomial-time algorithms Q1 , Q2 and Q3 must be available: Q1 , given I ∈ IΠ , produces a solution Q1 (I) ∈ SI ; Q2 , given an instance I ∈ IΠ and a binary vector x, determines whether x ∈ SI and, if so, computes c(x, I); Q3 , given an instance I ∈ IΠ and a solution x ∈ SI , either outputs y ∈ N (x, I) with c(y, I) < c(x, I) or states that there is no such y ∈ N (x, I), implying that x is locally optimal. A problem Π is PLS-reducible to a problem Π0 if there are two polynomial-time computable functions h : IΠ → IΠ0 and g : Sh(I) → SI such that: h maps any instance I of Π to an instance h(I) of Π0 ; g maps any solution y of h(I) to a solution g(y) of I and for all I ∈ IΠ , if y ∗ is a local optimum for h(I), then g(y ∗ ) is a local optimum for I. A problem Π in PLS is PLS-complete if every problem in PLS is PLS-reducible to Π. A well-known PLS-complete problem is POS NAE 3SAT [18], i.e. not-all-equal-3SAT with positive literals only: an instance consists of clauses in conjunctive normal form; each clause has at most three positive literals and is assigned a positive cost; each clause is satisfied if its constituents do not all have the same value. A solution is a truth assignment, i.e. a 0/1 assignment to all variables; the cost of a solution, to be minimized, is the sum of the costs of the satisfied clauses; the neighborhood of a solution contains all solutions obtained by flipping the value of one variable. The local search problem is to find a truth assignment whose cost cannot be decreased by flipping a variable.
3
Polynomial solvability of the symmetric case
In this section we investigate the computational complexity of symmetric TU congestion games. Theorem 1. There is a strongly polynomial-time algorithm for finding a pure Nash equilibrium in symmetric TU congestion games. Proof. Since TU matrices are closed under adding copies of rows and multiplying rows by −1, we can assume without loss of generality that the strategy set of every player i is given by: X i = {xi : Axi ≥ b, xi ∈ {0, 1}n }. The algorithm computes the global optimum of min{φ(x) : x ∈ X},
(4)
where φ is the potential function of the congestion game and X = X 1 × · · · × X N is the strategy space. Since a global optimum of (4) is also a local optimum of the local search problem, the resulting state is a pure Nash equilibrium. In the first phase of our algorithm, we set up an aggregated problem as follows. Since the value of the potential function (2) only depends on how many players use a resource, we sum up the constraints corresponding to a given row P of matrix A for all players, and we define the aggregated N variables z ∈ [0, N ]n ∩ Zn such that zj = i=1 xij for each resource j. zj n X X min dj (i) : Az ≥ N b, z ∈ [0, N ]n ∩ Zn . (5) j=1 i=1
In the remainder of the proof, we find an optimum z¯ of (5), and then we show how to decompose it into a state x ¯ ∈ X such that z¯ = x ¯1 + · · · + x ¯N , with x ¯i ∈ X i . Since, for each state x ∈ X, the 1 N corresponding z = x + · · · + x is feasible for (5) and has the same objective value, we have that x ¯ is an optimum of (4). Next, to model the objective function of (5) as a linear function, we introduce variables y i ∈ {0, 1}n for i = 1, . . . , N , where yji = 1 if at least i players use resource j, and yji = 0 otherwise. Since 5
z = y 1 + · · · + y N for each resource j, we can write our aggregated problem as: minimize
n X N X
dj (i)yji
(6)
j=1 i=1
subject to
N X
Ay i ≥ N b
i=1
0 ≤ yi ≤ 1 i
i = 1, . . . , N
n
y ∈ {0, 1}
i = 1, . . . , N.
First, for each z feasible for (5), we define yji = 1 if i ≤ zj and yji = 0 otherwise. Note that y is feasible for (6) and with the same objective value as z. Moreover, for each y feasible for (6), the vector z = y 1 + · · · + y N is feasible for (5) and has objective value not larger than that of y. Therefore, the optimal solution y¯ of (6) yields an optimal solution z¯ of (5). We now show how to solve problem (6). First, the constraint matrix of the aggregated problem (6) is also TU, since it is of the form ( A |
A ··· {z
N times
A ). }
As the right hand side of the system is integral, the linear relaxation of the feasible set of the aggregated problem has only integral vertices (see for example Theorem 19.3 in [19]). Thus we can find an optimal solution y¯ of the aggregated problem via linear programming. Using Tardos’s [22] algorithm, this can be done in time polynomial in size(A) and in N , thus in time polynomial in n, m, N because all entries of A are in {0, ±1}. The vector z¯ = y¯1 + . . . y¯N is then an optimum of (5). In the remainder of the proof, we show how to derive from z¯ a state x ¯ with the same objective value in (4). Since A¯ z ≥ N b, and 0 ≤ z¯j ≤ N for every j = 1, . . . , n, z¯ is an integral vector in N P , where P = {x : 0 ≤ x ≤ 1, Ax ≥ b}. Since A is TU, the polyhedron P has the integer decomposition property, thus there exist integer vectors x ¯1 , . . . , x ¯N in P such that z¯ = x ¯1 + · · · + x ¯N . Following Baum and Trotter [2], we show how to obtain such vectors in strongly polynomial time. We show how to find an integer vector x ¯1 in P such that z¯ − x ¯1 is an integer vector in (N − 1)P . In order to do so, we define P 1 = {s : 0 ≤ s ≤ 1, z¯ − (N − 1) ≤ s ≤ z¯, b ≤ As ≤ A¯ z − (N − 1)b}.
(7)
The polyhedron P 1 is nonempty since it contains z¯/N , and is integral since its constraint matrix is TU. Again using Tardos’s [22] algorithm, we can find a vertex x ¯1 of P 1 in time polynomial in n, m. By applying the above argument recursively N times, we obtain integral vectors x ¯1 , x ¯2 , . . . , x ¯N in P with z¯ = x ¯1 + · · · + x ¯N . Therefore, x ¯ is a pure Nash equilibrium. The total running time of the algorithm is polynomial in n, m, N .
3.1
Consequences for combinatorial problems
In this section we show that the combinatorial game N , and games M, EC, SS, VC on bipartite graphs are TU congestion games. We then explain, from a combinatorial point of view, when such games are symmetric. In such cases, Theorem 1 implies that one can find in strongly polynomial time a pure Nash equilibrium. Proposition 1. Congestion games N , and congestion games M, EC, SS, VC on bipartite graphs are TU congestion games.
6
Proof. Network congestion games are TU congestion games, since xi is the incidence vector of a dipath from ri to si in digraph D = (V, E) if and only if Axi = bi , xi ∈ {0, 1}E , where A is the V × E incidence matrix of D, and bi has entry −1 corresponding to node ri , entry +1 corresponding to node si , and all other entries are 0. M and EC on bipartite graphs are TU congestion games, since xi is the incidence vector of a i matching (resp. edge cover) in Gi = (V i , E i ) if and only if Ai xi ≤ 1 (resp. Ai xi ≥ 1), xi ∈ {0, 1}E , where Ai is the V i × E i incidence matrix of Gi . SS and VC on bipartite graphs are TU congestion games, since xi is the incidence vector of a stable set (resp. vertex cover) in Gi = (V i , E i ) if and only if Ai xi ≤ 1 (resp. Axi ≥ 1), xi ∈ {0, 1}Vi , where Ai is the E i × V i incidence matrix of Gi . Recall that a TU congestion game is symmetric if Ai = A and bi = b for every player i = 1, . . . , N . As a consequence, games N are symmetric if all players have the same origin ri = r and destination si = s. Games M, EC, SS, VC are symmetric if all players act on the same subgraph, i.e. Gi = G for all i = 1, . . . , N . Theorem 1 then directly implies the following corollary: Corollary 1. There is a strongly polynomial-time algorithm for finding a pure Nash equilibrium in the symmetric case of games N , and of games M, EC, SS, VC on bipartite graphs. Interestingly, the algorithm given in the proof of Theorem 1 has some nice combinatorial interpretations in the above combinatorial games. For the symmetric case of N , we recover the algorithm described in [8]. Given a digraph D = (V, E) and arc delay functions de , we construct a new digraph D0 = (V, E 0 ) by replacing each arc e ∈ E with N parallel arcs e1 , . . . , eN between the same nodes, with delays de (1), . . . , de (N ) and capacity 1. Solving problem (6) is equivalent to finding an integer flow F of value N from origin r to destination s of minimum cost in the digraph D0 . This is a ˜ be the subgraph of D obtained by minimum cost flow problem (see Ch. 12 in [20]). Now let D ¯ be the set deleting arc e ∈ E if no edge among e1 , . . . , eN is used by the flow F . Moreover, let E of arcs e ∈ E such that all arcs e1 , . . . , eN are used by the flow F . Finding an integer vector in (7) ˜ that contains all arcs in is then equivalent to finding a directed path from r to s in the digraph D ¯ In the case where there are no directed cycles of negative delay in D0 , this is equivalent to just E. ˜ finding a directed path from r to s in D. For the remaining combinatorial congestion games, we will need the following construction. Given bipartite graph G = (V, E) and profits pe (or delays de ), we construct a new bipartite graph G0 = (V, E 0 ) by replacing each edge e ∈ E with N parallel edges e1 , . . . , eN between the same nodes, with profits pe (1), . . . , pe (N ) (or delays de (1), . . . , de (N )). For games M, solving problem (6) is equivalent to finding a maximum profit subset F of E 0 such that each node v in V is incident to at most N edges in F . This is a simple b-matching problem (see ˜ be the subgraph of G obtained by deleting edge e ∈ E if no edge among Ch. 21 in [20]). Now let G 1 N e , . . . , e is in F . Moreover, let U be the set of nodes that have degree N in (V, F ). Finding an ˜ covering all nodes in U . integer vector in (7) is then equivalent to finding a matching in the graph G For games EC, solving problem (6) is equivalent to finding a minimum cost subset F of E 0 such that each node v in V is incident to at least N edges in F . This is a simple b-edge cover problem ˜ be the subgraph of G obtained by deleting edge e ∈ E if either no (see Ch. 21 in [20]). Now let G 1 N edge among e , . . . , e is in F , or if if all edges e1 , . . . , eN are in F . Finding an integer vector in (7) ˜ such that for every node v ∈ V , v is is then equivalent to finding a set of edges C in in the graph G incident to a number of edges in C between bounds lv and uv that can be easily calculated. For games VC and SS, one can find an integer vector x ¯1 ∈ P 1 by finding a perfect vertex cover, ˜ of G. For vertex cover congestion i.e. a vertex cover that is also a stable set, in a suitable subgraph G ˜ games, G is the subgraph of G obtained by first deleting nodes v ∈ V if z¯v = 0 (these nodes will have x ¯1v = 0), then by deleting all edges uv with z¯u + z¯v 6= N , and finally by removing the isolated nodes ˜ is the subgraph of G obtained (these nodes will have x ¯1v = 1). For stable set congestion games, G 7
by first deleting nodes v ∈ V if z¯v = N (these nodes will have x ¯1v = 1), then by deleting all edges uv with z¯u + z¯v 6= N , and finally by removing the isolated nodes (these nodes will have x ¯1v = 0).
4
More combinatorial games
In this section we consider some variants of the combinatorial games previously defined that do not appear to be TU congestion games, but where we can still find a pure Nash equilibrium in strongly polynomial time. A maximum cardinality matching congestion games (C − M) is a variant of M where each set X i consists only of the maximum cardinality matchings in Gi . Similarly, in minimum cardinality edge cover congestion games (C − EC) each set X i consists of the minimum cardinality edge covers in Gi ; In maximum cardinality stable set congestion games (C − SS) each set X i consists of the maximum cardinality stable sets in Gi ; In minimum cardinality vertex cover congestion games (C − VC) each set X i consists of the minimum cardinality vertex covers in Gi . Proposition 2. Given an instance I of a C − M (resp. C − EC, C − SS, C − VC) on a graph G, one can construct in strongly polynomial time an instance h(I) of a M (resp. EC, SS, VC) on the same graph G, so that a pure Nash equilibrium in h(I) is also a pure Nash equilibrium in I. Proof. We give the reduction for matching games, the other ones being analogous. To map an instance I of C − M to an instance h(I) of M, we only need to modify the edge profits. Let G = (V, E). Let pe (i), i = 1, . . . , N , be the given profits in I, and let ∆ = max{|pe (i)| : e ∈ E, i = 1, . . . , N }. We set the profits in h(I) to be p0e (i) = pe (i) + 2|E|∆ + 1. It can be checked that for every player i, the payoff in h(I) corresponding to a matching in Gi that is of maximum cardinality will always be strictly larger than a payoff in h(I) corresponding to a matching in Gi that is not of maximum cardinality. We claim that a pure Nash equilibrium x for h(I) is also a pure Nash equilibrium for I. By construction, the strategy of each player i in x corresponds to a maximum cardinality matching M i in Gi . The payoff f˜i (x) of player i in h(I) is f˜i (x) = f i (x) + k(2|E|∆ + 1), where f i (x) is the payoff of player i in I, and k is the cardinality of a maximum cardinality matching in Gi . By contradiction, assume that there is another maximum cardinality matching N i in Gi such that in the state x0 obtained from x by swapping M i with N i , player i has better payoff in I, i.e., f i (x0 ) > f i (x). This implies f˜i (x0 ) = f i (x0 ) + k(2|E|∆ + 1) > f i (x) + k(2|E|∆ + 1) = f˜i (x), contradicting the fact that x is a pure Nash equilibrium for h(I). The given reduction is strongly polynomial. By Proposition 2 and Corollary 1 we obtain the following: Corollary 2. There is a strongly polynomial-time algorithm for finding a pure Nash equilibrium in the symmetric case of the following games on bipartite graphs: C − M, C − EC, C − SS, C − VC.
5
PLS-completeness of the asymmetric case
In this section we focus on asymmetric TU congestion games, i.e. congestion games in their most general form (3). The next Proposition follows directly from PLS-completeness of asymmetric network congestion games [8], see proof of Proposition 1. Proposition 3. Asymmetric TU congestion games in the form (3) are PLS-complete, even if A = Ai , i = 1, . . . , N . We remark that, in fact, all the asymmetric variants of problems M, EC, SS and VC on bipartite graphs can be written as TU congestion games with A = Ai , for i = 1, . . . , N . Our goal is to prove that all these asymmetric TU congestion are PLS-complete. 8
To this purpose we define the perfect matching congestion game (PM) as a variant of M where for every player i, the subgraph Gi admits a perfect matching, and where the set X i consists only of the perfect matchings in Gi . PM is equivalent to (i) C − M where all subgraphs Gi admit a perfect matching; (ii) C − EC where all subgraphs Gi admit a perfect matching, and the edge delays are the negative of the profits. Similarly, the perfect vertex cover congestion game (PVC) is a variant of VC where for every player i, the subgraph Gi admits a perfect vertex cover, i.e., a vertex cover that is also a stable set, and where the set X i consists only of the perfect vertex covers in Gi . PVC is equivalent to (i) C − VC where all subgraphs Gi admit a perfect vertex cover; (ii) C − SS where all subgraphs Gi admit a perfect matching and the vertex profits are the negative of the delays. We first show that the asymmetric PM on a bipartite graph is PLS-complete. Theorem 2. It is PLS-complete to find a pure Nash equilibrium in the asymmetric perfect matching congestion game on a bipartite graph. Proof. We give a PLS-reduction of POS NAE 3SAT to an asymmetric PM on a bipartite graph G. First, we define a map h from any instance of POS NAE 3SAT to an instance of an asymmetric PM on a bipartite graph G. Given an instance I of POS NAE 3SAT, we construct a congestion game h(I) as follows. Denote by C = {c1 , . . . , cn } the clauses of I and by {x1 , . . . , xN } its variables. Let wj be the cost of clause cj and, for a truth assignment x of I denote by w(x) the cost of x. Each variable of POS NAE 3SAT is a player of the PM and each NAE clause is a set of resources, i.e. a set of edges. Precisely, for each NAE clause cj we build the graph “gadget” in Fig. 2a. The graph gadget of clause cj is a 4-cycle uj , vj , zj , v¯j , uj . vj+1
vj
vj +
2
vj zj
zj
uj
zj+1
uj+1
uj +
2
zj +
uj
2
v¯j
v¯j
v¯j+1
v¯j +
2
(a)
(b)
Figure 2: Reduction from POS NAE 3SAT to asymmetric PM on a bipartite graph. Let mj ∈ {1, 2, 3} denote the number of variables in cj . The edge profits are defined as follows: • If e = v¯j uj or e = v¯j zj , then pe (i) = 0 for i = 1, . . . , N ; • If e = uj vj and cj contains at least a constant equal to 1, then pe (i) = 0 for i = 1, . . . , N ; otherwise, pe (i) = 0 for i = 1, . . . , mj − 1 and pe (i) = −wj for i = mj , . . . , N ; • If e = vj zj and cj contains at least a constant equal to 0, then pe (i) = 0 for i = 1, . . . , N ; otherwise, pe (i) = 0 for i = 1, . . . , mj − 1 and pe (i) = −wj for i = mj , . . . , N . Now, we build graph G as follows: for any two clauses cj and cj+1 , j = 1, . . . , n−1 we identify zj and uj+1 ; moreover, we identify zn and u1 , see Fig. 2b. Clearly, G is a bipartite graph with bipartitions {vj , v¯j }j=1,...,n and {uj }j=1,...,n . For each i = 1, . . . , N , let C(i) denote the set of clauses containing variable xi and let Vi = {uj , zj , j = 1, . . . , n} ∪ {vj : cj ∈ C(i)} ∪ {¯ vj : cj ∈ / C(i)}. We assign to
9
player i the subgraph Gi of G induced by nodes in Vi . This shows how to map I to an instance h(I) of asymmetric PM on a bipartite graph. Next, we define a map g from states of h(I) to truth assignments of I. A state of h(I) is a set of N perfect matchings on graphs Gi , i = 1, . . . , N . Note that each subgraph Gi is a cycle of length 2n that admits two perfect matchings: M0i = {uj vj : cj ∈ C(i)} ∪ {uj v¯j : cj ∈ / C(i)} and vj zj : cj ∈ / C(i)}. Let gi : {M0i , M1i } → {0, 1} such that gi (M0i ) = 0 and M1i = {vj zj : cj ∈ C(i)} ∪ {¯ gi (M1i ) = 1. We map strategy M i ∈ {M0i , M1i } of player i to xi = gi (M i ). Setting g = (g1 , . . . , gN ) shows that any state of h(I) is mapped to a truth assignment of I, and that the mapping is bijective. Finally, we need to show that any pure Nash equilibrium of h(I) maps to a local minimum of I. First, we remark that, for any truth assignment x of I, each x0 ∈ N (x, I) obtained by flipping variable xi , i ∈ {1, . . . , N } is in one-to-one correspondence with the state obtained from g −1 (x) after the defection of player i. Now, let M 1 , . . . , M N be a a pure Nash equilibrium of h(I), where M i ∈ {M0i , M1i } for all i = 1, . . . , N , and denote by y the corresponding state. Then, for any state y 0 obtained from y by switching perfect matching M i , we have that fi (y) − fi (y 0 ) ≥ 0, where fi denotes the payoff of player i in PM. By construction, for x = g(y) and x0 = g(y 0 ) we have that w(x0 ) − w(x) = fi (y) − fi (y 0 ) ≥ 0. This proves that x is a local optimum of I. Theorem 2 directly implies the following result. Corollary 3. It is PLS-complete to find a pure Nash equilibrium (i) in the asymmetric matching congestion game on a bipartite graph; (ii) in the asymmetric edge cover congestion game on a bipartite graph. Proof. (i) Since PM is equivalent to C − M when all subgraphs Gi admit a perfect matching, we have that asymmetric C − M is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 2. (ii) Since PM is equivalent to C − EC when all subgraphs Gi admit a perfect matching, we have that asymmetric C − EC is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 2. We now show that the asymmetric PVC on a bipartite graph is PLS-complete. Theorem 3. It is PLS-complete to find a pure Nash equilibrium in the asymmetric perfect vertex cover congestion game on a bipartite graph. Proof. We give a PLS-reduction of POS NAE 3SAT to an asymmetric PVC on a bipartite graph G. The proof structure is similar to that of Theorem 3, thus here we only outline the main differences. Again, the variables of POS NAE 3SAT map to players of PVC and clauses map to a set of resources. For each NAE clause cj we build the graph “gadget” in Fig. 3a, that is a 8-cycle uj , sj , vj , tj , zj , t¯j , v¯j , s¯j , uj . For mj ∈ {1, 2, 3} the vertex profits are defined as: • If u ∈ / {sj , vj }, then du (i) = 0 for i = 1, . . . , N ; • If u = sj and cj contains at least a constant equal to 1, then du (i) = 0 for i = 1, . . . , N ; otherwise, du (i) = 0 for i = 1, . . . , mj − 1 and du (i) = wj for i = mj , . . . , N ; • If u = vj and cj contains at least a constant equal to 0, then du (i) = 0 for i = 1, . . . , N ; otherwise, du (i) = 0 for i = 1, . . . , mj − 1 and du (i) = wj for i = mj , . . . , N . We build a bipartite graph G by identifying zn and u1 and zj and uj+1 for j = 1, . . . , n − 1, see Fig. 3b. For each i = 1, . . . , N , we let Vi = {uj , zj , j = 1, . . . , n} ∪ {sj , vj , tj : cj ∈ C(i)} ∪ {¯ sj , v¯j , t¯j : cj ∈ / C(i)} and we define Gi as the subgraph of G induced by nodes in Vi . Since each Gi is a cycle of length 4n, it admits exactly two perfect vertex covers: W0i = {sj , tj : cj ∈ C(i)} ∪ {¯ sj , t¯j : cj ∈ / C(i)} and W1i = {uj , vj : cj ∈ C(i)} ∪ {uj , v¯j : cj ∈ / C(i)}. We define i i gi : {W0 , W1 } → {0, 1} such that gi (W0i ) = 0 and gi (W1i ) = 1 and we map strategy W i ∈ {W0i , W1i } of player i to xi = gi (W i ). Let y be a Nash equilibrium of PVC corresponding to vertex covers W 1 , . . . , W N , and denote by −fi the cost function of player i. Then, for any state y 0 obtained by switching perfect vertex cover 10
vj+1
vj
sj+1
tj
vj
uj
2
zj
zj t¯j
s¯j v¯j
2
tj +
sj
tj
sj
vj +
tj+1 sj +2 zj+1
uj+1
uj ¯tj
s¯j v¯j
uj +
2
zj +
s¯j +
s¯j+1
t¯j+1 v¯j+1
2
2
t¯j +
2
v¯j +
2
(a)
(b)
Figure 3: Reduction from POS NAE 3SAT to asymmetric PVC on a bipartite graph. W i , we have that fi (y 0 ) − fi (y) ≥ 0. By construction, for x = g(y) and x0 = g(y 0 ) it follows that w(x0 ) − w(x) = fi (y 0 ) − fi (y) ≥ 0. Theorem 3 directly implies the following result. Corollary 4. It is PLS-complete to find a pure Nash equilibrium (i) in the asymmetric vertex cover congestion game on a bipartite graph; (ii) in the asymmetric stable set congestion game on a bipartite graph. Proof. (i) Since PVC is equivalent to C − VC when all subgraphs Gi admit a perfect vertex cover, we have that asymmetric C − VC is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 3. (ii) Since PVC is equivalent to C − SS when all subgraphs Gi admit a perfect vertex cover, we have that asymmetric C − SS is PLS-complete on a bipartite graph. The result then follows by Proposition 2 and Theorem 3.
6
Conclusion
We have introduced the notion of TU congestion games and we have shown that finding a pure Nash equilibrium can be done in strongly polynomial time in the symmetric case. We have also demonstrated that, for several combinatorial variants of TU Congestion Games, the asymmetric case is PLS-complete. Our results suggest that the computational complexity of a game may depend on the complexity of a polyhedral representation of the strategy sets of players. A crucial property in the design of our algorithm is the integral decomposition property, allowing us to decompose an aggregated strategy that minimizes the potential function into a pure Nash equilibrium. This property may also hold if the constraint matrix of the symmetric TU congestion game is not TU. Open questions involve extensions of our results to non-TU congestion games and approximation results in the general case.
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