A note on partition into triangles parametrized by tree-width
A note on partition into triangles parametrized by tree-width∗ Dušan Knop1 1
Department of Applied Mathematics, Charles University, Prague, Czech Republic [email protected]
arXiv:1508.04725v1 [cs.DM] 19 Aug 2015
Abstract We study the parametrized complexity of the Partition into Triangles problem. For this problem a (simple) graph with 3n vertices is given and the question is whether it is possible to cover its vertices with n triangles (complete graphs on 3 vertices). We prove that there is an FPT algorithm that decides the Partition into Triangles problem and that the existence of a polynomial size kernel is unlikely (unless NP ⊆ coNP/poly). 1998 ACM Subject Classification G.2.2 Graph Theory Keywords and phrases triangle partition, tree-width, parameterized algorithm
1
Introduction
We begin this section with several definitions needed to define the Partition into Triangles problem. We then proceed with an overview of previous result concerning the Partition into Triangles problem and related articles. Finally, we enclose this section with our results.
1.1
Problem definition
For a set U we denote by U2 the set of all subsets of U having size 2, that is U2 := {{u, v} : u, v ∈ U and u 6= v}. Further, by 2U we denote the set of all subsets of the set U, that is {S : S ⊆ U }. Let G = (V, E) and H = (W, F ) be graphs, we say that G is isomorphic to H (we write G ' H) if there exists a bijective mapping f : V → W such that {u, v} ∈ E if and only if {f (u), f (v)} ∈ F. A clique on n vertices is a graph with vertex set V = {1, 2, . . . , n} and edge set V2 , we denote this graph as Kn . For more notation on graphs, we refer reader to a monograph by Diestel [4]. We say that a graph G = (V, E) with 3n vertices has a partition into triangles if there is a partition V1 ∪ V2 ∪ · · · ∪ Vn of the set V, such that |Vi | = 3 for every i = 1, 2, . . . , n and G[Vi ] ' K3 . I Problem 1 (Partition into Triangles). INPUT: QUESTION:
∗
A graph G = (V, E) with |V | = 3n, for some (positive) integer n. Is there a partition into triangles for G?
Research supported by the CE-ITI grant project P202/12/G061 of GA ČR, grant project Kontakt LH 12095, by GAUK project 1784214 and by the project SVV–2015–260223. © Dušan Knop; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
2
A note on partition into triangles
1.2
Previous results
The Partition into Triangles problem arises as a special case of the Set Partition problem, which is defined as follows. I Problem 2 (Set Partition). INPUT: QUESTION:
A set (universe) U and a set system {S1 , S2 , . . . , Sn } ⊆ 2U . Sk Are there disjoint sets Si(1) , Si(2) , . . . , Si(k) such that j=1 Si(j) = U.
The Set Partition problem is well-known to be NP-complete [7], while it is easy to show that Partition into Triangles problem is NP-complete (for example by a reduction from the 3D-Matching problem—the problem of finding a perfect matching in 3-uniform 3-partite hypergraph).
1.2.1
Exact exponential algorithms
The Partition into Triangles problem is well studied from the point of view of exact exponential algorithms. Here we write O∗ (cn ) when there exists a constant d such that the overall running time is O(cn nd ). It is possible to use general inclusion-exclusion [2] formula to obtain an algorithm with running time O∗ (2n ) and polynomial space. The best algorithm for the Partition into Triangles problem we are aware of is by Björklund [1] and runs in time O∗ (1.496n ) and polynomial space, but this algorithm is probabilistic with one-sided error. Another approach in exact exponential algorithms for the Partition into Triangles problem is by Bodlaender et al. [12]. They study the problem on graphs of bounded degree and prove that the problem is polynomially solvable on graphs with maximum 3, while it is NP-hard on graph with maximum degree 4. Moreover for the case of graph with maximum degree 4, they show an algorithm running in time O∗ (1.0222n ) and polynomial space. Koivisto [10] analyzed the time complexity of the Set Partition problem when partition into sets of bounded cardinality. His approach leads to an O∗ (1.7693n ) for the Partition into Triangles problem. In this particular case the running time can be further improved to O∗ (1.7549n ) as showed by Kloks [9] by more careful analysis of the previous result by Koivisto.
1.2.2
Parameterized complexity
From the point of view of parametrized complexity, we are not aware on any previous result with respect to a stuctural graph parameters (such as tree-width of clique width). There is trivial and easy to see FPT algorithm for graphs of bounded vertex cover number. For these graphs the parameter must be of order O(|V (G)|) and thus it suffices to apply an exact exponential algorithm. As pointed out by Gajarský et al. [6] the parametrized complexity of the Partition into Triangles problem is unknown on graphs of bounded tree-width.
1.3
Our contribution
In this paper we resolve the fixed parameterized complexity of the Partition into Triangles problem parameterized by the tree-width of the underlying graph. The result is summarized in the following theorem: I Theorem 3. For a graph G with a given tree decomposition of tree-width w, it is possible to decide the Partition into Triangles problem in time O((2w )2 · n).
D. Knop
3
We also refute the existence of polynomial kernel, when parameterized by a stronger structural graph parameter—path-width: I Theorem 4. There is no polynomial kernel for the Partition into Triangles problem parameterized by the path-width of the underlying graph, unless NP ⊆ coNP/poly.
2
Preliminaries
In this section we review some known facts from tree-width (Subsection 2.1) and techniques for refuting polynomial kernels (Subsection 2.2).
2.1
Preliminaries on tree-width
In this section we cover basic facts about tree-width to make the rest of the paper easier to understand. I Definition 5 (Tree decomposition [11]). Let G = (V, E) be a graph. A tree decomposition of G is a tree T = (B, F ) such that: for every B ∈ B it holds that B ⊆ V, for every v ∈ V there exists B ∈ B such that v ∈ B, for every e ∈ E there exists B ∈ B such that e ⊆ B, for every v ∈ V the graph induced by the set {B ∈ B : v ∈ B} is connected in T. The width of the decomposition T is the max{|B| : B ∈ B}. We call a set B ∈ B a bag. The tree-width of a graph G, denoted as tw(G) is the minimum width of any tree decomposition of the graph G. When the tree T happens to be a path, we speak of path decomposition and further define as path-width of a graph G, denoted as pw(G), the minimum width of any path decomposition of the graph G.
2.1.0.1
Nice tree decomposition
When designing an algorithm, one usually works with nice tree decomposition. We say T = (B, F ) is a nice tree decomposition if it is a tree decomposition and it further holds that the tree T is rooted at a bag R, we say B 0 is a child bag of a bag B if {B, B 0 } ∈ F and dT (B 0 , R) = dT (B, R)—that is B is right before B 0 on the (unique) path from B 0 to the root R— if Bis on the path from B 0 to the root R then we say that B 0 is a descendant bag of the bag B, every leaf of the tree T is a singleton (a set with only one element), every bag has at most two children, if a bag B has two children B1 , B2 it holds that B = B1 = B2 , in this case we say this is a join bag, if a bag B has only one child B 0 it holds that |B∆B 0 | = 1, here if |B \ B 0 | = 1 then we say that B is an introduce bag, otherwise we say that B is a forget bag. I Proposition 6 ([8]). Let a tree decomposition T be given for a graph G, it is po For a graph G with a nice tree decomposition T = (B, F ) and a bag B ∈ B we define a subgraph associated with a bag B, denote it as GB , as a graph induced by the set vertices {v ∈ B 0 : B 0 is a descendant of B}.
4
A note on partition into triangles
2.2
Preliminaries on refuting polynomial kernels
Here we present simplified review of a framework used to refute existence of polynomial kernel for a parameterized problem from Chapter 15 of a monograph by Cygan et al. [3]. In the following we denote by Σ a final alphabet, by Σ∗ we denote the set of all words over Σ and by Σ≤n we denote the set of all words over Σ and length at most n. I Definition 7 (Polynomial equivalence relation). An equivalence relation R on the set Σ∗ is called polynomial equivalence relation if the following conditions are satisfied: 1. there exists an algorithm that, given strings x, y ∈ Σ∗ , resolves whether x ≡R y in time polynomial in |x| + |y|, and 2. relation R restricted to the set Σ≤n has at most p(n) equivalence classes, for some polynomial p(·). I Definition 8 (Cross-composition). Let L ⊆ Σ∗ be an unparameterized language and Q ⊆ Σ∗ × N be a parametrized language. We say that L cross-composes into Q if there exists a polynomial equivalence relation R and an algorithm A, called the cross-composition, satisfying the following conditions. The algorithm A takes on input a sequence of strings Pt x1 , x2 , . . . , xt ∈ Σ∗ that are equivalent with respect to R,runs in polynomial time in i=1 |xi |, and outputs one instance (y, k) ∈ Σ × N such that: 1. k ≤ p(maxti=1 |xi |, log t) for some polynomial p(·, ·), and 2. (y, k) ∈ Q if and only if xi ∈ L for all i. With this framework, it is possible to refute even stronger reduction techniques—namely polynomial compression—according to the following definition: I Definition 9 (Polynomial compression). A polynomial compression of a parameterized language Q ⊆ Σ∗ × N into an unparameterized language R ⊆ Σ∗ is an algorithm that takes as input an instance (y, k) ∈ Σ∗ × N, works in polynomial time in |x| + k, and returns a string y such that: 1. |y| ≤ p(k) for some polynomial p(·), and 2. y ∈ R if and only if (x, k) ∈ Q. It is possible to refute existence of polynomial kernel using Definitions 7,8 and 9 with the help of use of the following theorems and a complexity assumption that is unlikely to hold—namely NP ⊆coNP/poly. I Theorem 10 ([5]). Let L, R ⊆ Σ∗ be two languages. Assume that there exists an ANDdistillation of L into R. Then L ∈ coNP/poly. I Theorem 11. Assume that an NP-hard language L AND-cross-composes to a parameterized language Q. Then Q does not admit a polynomial compression, unless NP ⊆coNP/poly.
3
Partition into Triangles parametrized by tree-width
In this section, we show how to use a dynamic programming approach to prove that Partition into Triangles is fixed parameter tractable when parametrized by tree-width (Subsection 3.1). Moreover, we show that existence of polynomial kernel for the Partition into Triangles problem parameterized by tree-width is unlikely (Subsection 3.2).
D. Knop
3.1
5
FPT algorithm for the Partition into Triangles problem
Dynamic programming table As usual, we will store an information tableau at every bag of the tree decomposition. In a tableau there will be an entry for every subset of the vertices of the current bag and the information stored will just tell whether this is realizable subset or not. At every step the algorithm will compute what are the possible positions of the vertices of the bag X that partitions the graph GX but may be with the exception of vertices in X.
The description of the algorithm Our algorithm uses nice tree-decompositions and so it suffices to give the description for the respective types of bags. In the following X is the set of vertices of the current bag. leaf {x}—∅ is realizable, while the singleton set {x} is not. forget bag of {x}—a set A is realizable if the set A ∪ {x} was realizable. introduce bag of {x}—if a set A does not contain x that it is realizable if it was, while the set A ∪ {x} is realizable if there exist vertices y, z ∈ A such that x, y, z forms a triangle in the graph GX and the set A \ {y, z} was realizable. join bag—a set A is realizable if it can be decomposed to sets A1 , A2 such that A is the disjoint union of these sets and A1 was realizable in one child bag, while A2 was realizable in the other child bag. Now it is easy to see that it is possible to cover a graph G with triangles if and only if the set X is realizable for the root bag of the tree decomposition.
Running time analysis Here by w we denote the width of the tree decomposition given to the algorithm. leaf bag: trivially O(1), forget bag: trivially O(2w ), introduce bag: O(2w w2 ) as there are O(w2 ) pairs of adjacent vertices {y, z} in the bag, join bag: O((2w )2 ) as for a fixed set A we have to check all sets A0 in the left child and their complements in the right child, as this can be done in constant time, the claim follows. This finishes the proof of Theorem 3.
3.2
Existence of polynomial kernel is unlikely
In this section, we prove that the Partition into Triangles cross-composes into itself parameterized by treewidth. This implies Theorem 4 by the use of framework of Section 2.2.
3.2.0.2
Polynomial equivalence relation R
We can simply say that graphs G, H are equivalent with respect to R when |V (G)| = |V (H)|. It is easy to check that this is polynomial equivalence relation.
3.2.0.3
Cross-composition
Let G1 , G2 , . . . , Gt be input graphs for the Partition into Triangles problem equivalent with respect to R. The cross-composition is a graph G defined as the disjoint union of graphs G1 , G2 , . . . , Gt . Now, it suffices to check that:
6
A note on partition into triangles
Pt 1. it is possible to construct G in time polynomial in i=1 (|V (Gi )| + |E(Gi )|), 2. pw(G) ≤ maxti=1 |V (Gi )|, and as usually tw(G) ≤ pw(G), and 3. it is possible to partition G into triangles if and only if it is possible to partition each graph Gi into triangles. As parts 1 and 3 are straightforward, we proof only part 2 here. For this we just put every graph Gi into its own bag and connect these bags into a path. Note that this works even if input graphs are not equivalent with respect to R. Now the results of Section 2.2 implies that an existence of polynomial compression is unlikely with respect to path-width as a parameter and thus the Theorem 4 follows. References 1
2 3 4 5
6
7 8 9 10
11 12
Andreas Björklund. Exact Covers via Determinants. In Jean-Yves Marion and Thomas Schwentick, editors, 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science, pages 95–106, Nancy, France, 2010. Inria Nancy Grand Est & Loria. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusionexclusion. SIAM Journal on Computing, 39(2):546–563, 2009. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Andrew Drucker. New limits to classical and quantum instance compression. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 609–618, 2012. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, pages 163–176, 2013. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1979. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. Ton Kloks. A note on triangle partitions. ArXiv e-prints, March 2012. Mikko Koivisto. Partitioning into sets of bounded cardinality. In Jianer Chen and FedorV. Fomin, editors, Parameterized and Exact Computation, volume 5917 of Lecture Notes in Computer Science, pages 258–263. Springer Berlin Heidelberg, 2009. Neil Robertson and Paul D. Seymour. Graph minors. ii. algorithmic aspects of tree-width. Journal of Algorithms, 7(3):309 – 322, 1986. Johan M. M. van Rooij, Marcel E. van Kooten Niekerk, and Hans L. Bodlaender. Partition into triangles on bounded degree graphs. Theory of Computing Systems, 52(4):687–718, 2013.
Department of Applied Mathematics, Charles University, Prague, Czech Republic [email protected]
arXiv:1508.04725v1 [cs.DM] 19 Aug 2015
Abstract We study the parametrized complexity of the Partition into Triangles problem. For this problem a (simple) graph with 3n vertices is given and the question is whether it is possible to cover its vertices with n triangles (complete graphs on 3 vertices). We prove that there is an FPT algorithm that decides the Partition into Triangles problem and that the existence of a polynomial size kernel is unlikely (unless NP ⊆ coNP/poly). 1998 ACM Subject Classification G.2.2 Graph Theory Keywords and phrases triangle partition, tree-width, parameterized algorithm
1
Introduction
We begin this section with several definitions needed to define the Partition into Triangles problem. We then proceed with an overview of previous result concerning the Partition into Triangles problem and related articles. Finally, we enclose this section with our results.
1.1
Problem definition
For a set U we denote by U2 the set of all subsets of U having size 2, that is U2 := {{u, v} : u, v ∈ U and u 6= v}. Further, by 2U we denote the set of all subsets of the set U, that is {S : S ⊆ U }. Let G = (V, E) and H = (W, F ) be graphs, we say that G is isomorphic to H (we write G ' H) if there exists a bijective mapping f : V → W such that {u, v} ∈ E if and only if {f (u), f (v)} ∈ F. A clique on n vertices is a graph with vertex set V = {1, 2, . . . , n} and edge set V2 , we denote this graph as Kn . For more notation on graphs, we refer reader to a monograph by Diestel [4]. We say that a graph G = (V, E) with 3n vertices has a partition into triangles if there is a partition V1 ∪ V2 ∪ · · · ∪ Vn of the set V, such that |Vi | = 3 for every i = 1, 2, . . . , n and G[Vi ] ' K3 . I Problem 1 (Partition into Triangles). INPUT: QUESTION:
∗
A graph G = (V, E) with |V | = 3n, for some (positive) integer n. Is there a partition into triangles for G?
Research supported by the CE-ITI grant project P202/12/G061 of GA ČR, grant project Kontakt LH 12095, by GAUK project 1784214 and by the project SVV–2015–260223. © Dušan Knop; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
2
A note on partition into triangles
1.2
Previous results
The Partition into Triangles problem arises as a special case of the Set Partition problem, which is defined as follows. I Problem 2 (Set Partition). INPUT: QUESTION:
A set (universe) U and a set system {S1 , S2 , . . . , Sn } ⊆ 2U . Sk Are there disjoint sets Si(1) , Si(2) , . . . , Si(k) such that j=1 Si(j) = U.
The Set Partition problem is well-known to be NP-complete [7], while it is easy to show that Partition into Triangles problem is NP-complete (for example by a reduction from the 3D-Matching problem—the problem of finding a perfect matching in 3-uniform 3-partite hypergraph).
1.2.1
Exact exponential algorithms
The Partition into Triangles problem is well studied from the point of view of exact exponential algorithms. Here we write O∗ (cn ) when there exists a constant d such that the overall running time is O(cn nd ). It is possible to use general inclusion-exclusion [2] formula to obtain an algorithm with running time O∗ (2n ) and polynomial space. The best algorithm for the Partition into Triangles problem we are aware of is by Björklund [1] and runs in time O∗ (1.496n ) and polynomial space, but this algorithm is probabilistic with one-sided error. Another approach in exact exponential algorithms for the Partition into Triangles problem is by Bodlaender et al. [12]. They study the problem on graphs of bounded degree and prove that the problem is polynomially solvable on graphs with maximum 3, while it is NP-hard on graph with maximum degree 4. Moreover for the case of graph with maximum degree 4, they show an algorithm running in time O∗ (1.0222n ) and polynomial space. Koivisto [10] analyzed the time complexity of the Set Partition problem when partition into sets of bounded cardinality. His approach leads to an O∗ (1.7693n ) for the Partition into Triangles problem. In this particular case the running time can be further improved to O∗ (1.7549n ) as showed by Kloks [9] by more careful analysis of the previous result by Koivisto.
1.2.2
Parameterized complexity
From the point of view of parametrized complexity, we are not aware on any previous result with respect to a stuctural graph parameters (such as tree-width of clique width). There is trivial and easy to see FPT algorithm for graphs of bounded vertex cover number. For these graphs the parameter must be of order O(|V (G)|) and thus it suffices to apply an exact exponential algorithm. As pointed out by Gajarský et al. [6] the parametrized complexity of the Partition into Triangles problem is unknown on graphs of bounded tree-width.
1.3
Our contribution
In this paper we resolve the fixed parameterized complexity of the Partition into Triangles problem parameterized by the tree-width of the underlying graph. The result is summarized in the following theorem: I Theorem 3. For a graph G with a given tree decomposition of tree-width w, it is possible to decide the Partition into Triangles problem in time O((2w )2 · n).
D. Knop
3
We also refute the existence of polynomial kernel, when parameterized by a stronger structural graph parameter—path-width: I Theorem 4. There is no polynomial kernel for the Partition into Triangles problem parameterized by the path-width of the underlying graph, unless NP ⊆ coNP/poly.
2
Preliminaries
In this section we review some known facts from tree-width (Subsection 2.1) and techniques for refuting polynomial kernels (Subsection 2.2).
2.1
Preliminaries on tree-width
In this section we cover basic facts about tree-width to make the rest of the paper easier to understand. I Definition 5 (Tree decomposition [11]). Let G = (V, E) be a graph. A tree decomposition of G is a tree T = (B, F ) such that: for every B ∈ B it holds that B ⊆ V, for every v ∈ V there exists B ∈ B such that v ∈ B, for every e ∈ E there exists B ∈ B such that e ⊆ B, for every v ∈ V the graph induced by the set {B ∈ B : v ∈ B} is connected in T. The width of the decomposition T is the max{|B| : B ∈ B}. We call a set B ∈ B a bag. The tree-width of a graph G, denoted as tw(G) is the minimum width of any tree decomposition of the graph G. When the tree T happens to be a path, we speak of path decomposition and further define as path-width of a graph G, denoted as pw(G), the minimum width of any path decomposition of the graph G.
2.1.0.1
Nice tree decomposition
When designing an algorithm, one usually works with nice tree decomposition. We say T = (B, F ) is a nice tree decomposition if it is a tree decomposition and it further holds that the tree T is rooted at a bag R, we say B 0 is a child bag of a bag B if {B, B 0 } ∈ F and dT (B 0 , R) = dT (B, R)—that is B is right before B 0 on the (unique) path from B 0 to the root R— if Bis on the path from B 0 to the root R then we say that B 0 is a descendant bag of the bag B, every leaf of the tree T is a singleton (a set with only one element), every bag has at most two children, if a bag B has two children B1 , B2 it holds that B = B1 = B2 , in this case we say this is a join bag, if a bag B has only one child B 0 it holds that |B∆B 0 | = 1, here if |B \ B 0 | = 1 then we say that B is an introduce bag, otherwise we say that B is a forget bag. I Proposition 6 ([8]). Let a tree decomposition T be given for a graph G, it is po For a graph G with a nice tree decomposition T = (B, F ) and a bag B ∈ B we define a subgraph associated with a bag B, denote it as GB , as a graph induced by the set vertices {v ∈ B 0 : B 0 is a descendant of B}.
4
A note on partition into triangles
2.2
Preliminaries on refuting polynomial kernels
Here we present simplified review of a framework used to refute existence of polynomial kernel for a parameterized problem from Chapter 15 of a monograph by Cygan et al. [3]. In the following we denote by Σ a final alphabet, by Σ∗ we denote the set of all words over Σ and by Σ≤n we denote the set of all words over Σ and length at most n. I Definition 7 (Polynomial equivalence relation). An equivalence relation R on the set Σ∗ is called polynomial equivalence relation if the following conditions are satisfied: 1. there exists an algorithm that, given strings x, y ∈ Σ∗ , resolves whether x ≡R y in time polynomial in |x| + |y|, and 2. relation R restricted to the set Σ≤n has at most p(n) equivalence classes, for some polynomial p(·). I Definition 8 (Cross-composition). Let L ⊆ Σ∗ be an unparameterized language and Q ⊆ Σ∗ × N be a parametrized language. We say that L cross-composes into Q if there exists a polynomial equivalence relation R and an algorithm A, called the cross-composition, satisfying the following conditions. The algorithm A takes on input a sequence of strings Pt x1 , x2 , . . . , xt ∈ Σ∗ that are equivalent with respect to R,runs in polynomial time in i=1 |xi |, and outputs one instance (y, k) ∈ Σ × N such that: 1. k ≤ p(maxti=1 |xi |, log t) for some polynomial p(·, ·), and 2. (y, k) ∈ Q if and only if xi ∈ L for all i. With this framework, it is possible to refute even stronger reduction techniques—namely polynomial compression—according to the following definition: I Definition 9 (Polynomial compression). A polynomial compression of a parameterized language Q ⊆ Σ∗ × N into an unparameterized language R ⊆ Σ∗ is an algorithm that takes as input an instance (y, k) ∈ Σ∗ × N, works in polynomial time in |x| + k, and returns a string y such that: 1. |y| ≤ p(k) for some polynomial p(·), and 2. y ∈ R if and only if (x, k) ∈ Q. It is possible to refute existence of polynomial kernel using Definitions 7,8 and 9 with the help of use of the following theorems and a complexity assumption that is unlikely to hold—namely NP ⊆coNP/poly. I Theorem 10 ([5]). Let L, R ⊆ Σ∗ be two languages. Assume that there exists an ANDdistillation of L into R. Then L ∈ coNP/poly. I Theorem 11. Assume that an NP-hard language L AND-cross-composes to a parameterized language Q. Then Q does not admit a polynomial compression, unless NP ⊆coNP/poly.
3
Partition into Triangles parametrized by tree-width
In this section, we show how to use a dynamic programming approach to prove that Partition into Triangles is fixed parameter tractable when parametrized by tree-width (Subsection 3.1). Moreover, we show that existence of polynomial kernel for the Partition into Triangles problem parameterized by tree-width is unlikely (Subsection 3.2).
D. Knop
3.1
5
FPT algorithm for the Partition into Triangles problem
Dynamic programming table As usual, we will store an information tableau at every bag of the tree decomposition. In a tableau there will be an entry for every subset of the vertices of the current bag and the information stored will just tell whether this is realizable subset or not. At every step the algorithm will compute what are the possible positions of the vertices of the bag X that partitions the graph GX but may be with the exception of vertices in X.
The description of the algorithm Our algorithm uses nice tree-decompositions and so it suffices to give the description for the respective types of bags. In the following X is the set of vertices of the current bag. leaf {x}—∅ is realizable, while the singleton set {x} is not. forget bag of {x}—a set A is realizable if the set A ∪ {x} was realizable. introduce bag of {x}—if a set A does not contain x that it is realizable if it was, while the set A ∪ {x} is realizable if there exist vertices y, z ∈ A such that x, y, z forms a triangle in the graph GX and the set A \ {y, z} was realizable. join bag—a set A is realizable if it can be decomposed to sets A1 , A2 such that A is the disjoint union of these sets and A1 was realizable in one child bag, while A2 was realizable in the other child bag. Now it is easy to see that it is possible to cover a graph G with triangles if and only if the set X is realizable for the root bag of the tree decomposition.
Running time analysis Here by w we denote the width of the tree decomposition given to the algorithm. leaf bag: trivially O(1), forget bag: trivially O(2w ), introduce bag: O(2w w2 ) as there are O(w2 ) pairs of adjacent vertices {y, z} in the bag, join bag: O((2w )2 ) as for a fixed set A we have to check all sets A0 in the left child and their complements in the right child, as this can be done in constant time, the claim follows. This finishes the proof of Theorem 3.
3.2
Existence of polynomial kernel is unlikely
In this section, we prove that the Partition into Triangles cross-composes into itself parameterized by treewidth. This implies Theorem 4 by the use of framework of Section 2.2.
3.2.0.2
Polynomial equivalence relation R
We can simply say that graphs G, H are equivalent with respect to R when |V (G)| = |V (H)|. It is easy to check that this is polynomial equivalence relation.
3.2.0.3
Cross-composition
Let G1 , G2 , . . . , Gt be input graphs for the Partition into Triangles problem equivalent with respect to R. The cross-composition is a graph G defined as the disjoint union of graphs G1 , G2 , . . . , Gt . Now, it suffices to check that:
6
A note on partition into triangles
Pt 1. it is possible to construct G in time polynomial in i=1 (|V (Gi )| + |E(Gi )|), 2. pw(G) ≤ maxti=1 |V (Gi )|, and as usually tw(G) ≤ pw(G), and 3. it is possible to partition G into triangles if and only if it is possible to partition each graph Gi into triangles. As parts 1 and 3 are straightforward, we proof only part 2 here. For this we just put every graph Gi into its own bag and connect these bags into a path. Note that this works even if input graphs are not equivalent with respect to R. Now the results of Section 2.2 implies that an existence of polynomial compression is unlikely with respect to path-width as a parameter and thus the Theorem 4 follows. References 1
2 3 4 5
6
7 8 9 10
11 12
Andreas Björklund. Exact Covers via Determinants. In Jean-Yves Marion and Thomas Schwentick, editors, 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science, pages 95–106, Nancy, France, 2010. Inria Nancy Grand Est & Loria. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusionexclusion. SIAM Journal on Computing, 39(2):546–563, 2009. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Andrew Drucker. New limits to classical and quantum instance compression. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 609–618, 2012. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, pages 163–176, 2013. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1979. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. Ton Kloks. A note on triangle partitions. ArXiv e-prints, March 2012. Mikko Koivisto. Partitioning into sets of bounded cardinality. In Jianer Chen and FedorV. Fomin, editors, Parameterized and Exact Computation, volume 5917 of Lecture Notes in Computer Science, pages 258–263. Springer Berlin Heidelberg, 2009. Neil Robertson and Paul D. Seymour. Graph minors. ii. algorithmic aspects of tree-width. Journal of Algorithms, 7(3):309 – 322, 1986. Johan M. M. van Rooij, Marcel E. van Kooten Niekerk, and Hans L. Bodlaender. Partition into triangles on bounded degree graphs. Theory of Computing Systems, 52(4):687–718, 2013.
