Approximation Algorithms for Minimum Chain ... - Semantic Scholar
Approximation Algorithms for Minimum Chain Vertex Deletion
Sounaka Mishra
1
Mrinal Kumar1 Joint Work With N. Safina Devi 1
Saket Saurabh2
Indian Institute of Technology Madras, Chennai, India.1 The Institute of Mathematical Sciences, Chennai, India.2
19th February, 2011
Chain Graph I
Chain Graph: A bipartite graph G = (A ∪ B, E ) is called a chain graph if there exists an ordering ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } such that N(v1 ) ⊆ N(v2 ) . . . ⊆ N(vn ).
Chain Graph I
Chain Graph: A bipartite graph G = (A ∪ B, E ) is called a chain graph if there exists an ordering ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } such that N(v1 ) ⊆ N(v2 ) . . . ⊆ N(vn ).
I
Example: a
x
b
y
c
z
N (c) ⊆ N (b) ⊆ N (a) N (x) ⊆ N (y) ⊆ N (z)
Chain Graph I
Chain Graph: A bipartite graph G = (A ∪ B, E ) is called a chain graph if there exists an ordering ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } such that N(v1 ) ⊆ N(v2 ) . . . ⊆ N(vn ).
I
Example: a
x
b
y
c
z
N (c) ⊆ N (b) ⊆ N (a) N (x) ⊆ N (y) ⊆ N (z)
I
Observation : If G = (A ∪ B, E ) is a chain graph and ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } then vertices in B has a similar nested ordering.
Characterizations of Chain Graph
I
Theorem-1: [Yannakakis 1981] For a bipartite graph G = (A ∪ B, E ) the following are equivalent (a) G is a chain graph, (b) G is 2K2 -free, (c) C (G ) is chordal, (d) The bipartite complement of G is a chain graph.
Characterizations of Chain Graph
I
Theorem-1: [Yannakakis 1981] For a bipartite graph G = (A ∪ B, E ) the following are equivalent (a) G is a chain graph, (b) G is 2K2 -free, (c) C (G ) is chordal, (d) The bipartite complement of G is a chain graph.
I
Corollary-1: Given a bipartite graph G , it can be tested in polynomial time whether G is chain.
Characterizations of Chain Graph
I
Theorem-1: [Yannakakis 1981] For a bipartite graph G = (A ∪ B, E ) the following are equivalent (a) G is a chain graph, (b) G is 2K2 -free, (c) C (G ) is chordal, (d) The bipartite complement of G is a chain graph.
I
Corollary-1: Given a bipartite graph G , it can be tested in polynomial time whether G is chain.
I
For all pair of edges (u, v ) and (p, q) in G , test whether G [{u, v , p, q}] is isomerphic to 2K2 .
Minimum Chain vertex Deletion
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Minimum Chain Vertex Deletion: Given a bipartite graph G = (V , E ) and a weight function w : V → Q+ , find a set S ⊆ V P of minimum weight w (S) = v ∈S w (v ) such that G [V \ S] is a chain graph.
Minimum Chain vertex Deletion
I
Minimum Chain Vertex Deletion: Given a bipartite graph G = (V , E ) and a weight function w : V → Q+ , find a set S ⊆ V P of minimum weight w (S) = v ∈S w (v ) such that G [V \ S] is a chain graph.
I
Theorem-2 [Yannakakis 1981] Minimum Chain Vertex Deletion is NP-complete.
Minimum Chain vertex Deletion
I
Minimum Chain Vertex Deletion: Given a bipartite graph G = (V , E ) and a weight function w : V → Q+ , find a set S ⊆ V P of minimum weight w (S) = v ∈S w (v ) such that G [V \ S] is a chain graph.
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Theorem-2 [Yannakakis 1981] Minimum Chain Vertex Deletion is NP-complete.
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Minimum Chain Vertex Deletion can be approximated within a factor 4.
Minimum Chain vertex Deletion I
ILP formulation of Minimum Chain Vertex Deletion Given a graph G = (V , E ), let {T1 , T2 , . . . Tk } be the set of all 2K2 s in G . We assume that ∀ v ∈ V there is a Ti containing v . P ILP(G , w ) : Minimize w (v )xv P v ∈V Subject to v ∈Tj xv ≥ 1, 1 ≤ j ≤ k xv ∈ {0, 1},
∀v ∈ V .
Minimum Chain vertex Deletion I
ILP formulation of Minimum Chain Vertex Deletion Given a graph G = (V , E ), let {T1 , T2 , . . . Tk } be the set of all 2K2 s in G . We assume that ∀ v ∈ V there is a Ti containing v . P ILP(G , w ) : Minimize w (v )xv P v ∈V Subject to v ∈Tj xv ≥ 1, 1 ≤ j ≤ k xv ∈ {0, 1},
I
∀v ∈ V .
LP relaxation of ILP LP(G , w ) : Minimize Subject to
P
P v ∈V
v ∈Tj
w (v )xv xv ≥ 1, 1 ≤ j ≤ k
xv ≥ 0,
∀v ∈ V .
Minimum Chain vertex Deletion
I
Lemma-1: Let x be an optimal solution of LP(G , w ) for a given bipartite graph G = (A ∪ B, E ). If xv < 13 , for all v ∈ A ∪ B, then one can compute another optimal solution x 0 for LP(G , w ) with x 0 (v ) ≥ 31 , for some v ∈ A ∪ B.
Factor 3 Approximation Algorithm Algorithm 1 Input: A bipartite graph G = (A ∪ B, E ) and w : A ∪ B → Q+ . Output: A chain deletion set S for G . S = φ; while G has at least one 2K2 do { Construct LP(G , w ) and compute an optimal solution x satisfying the property in Lemma-1; compute the sets P = {v |xv ≥ 31 }, Z = {v |xv = 0}; S = S ∪ P; G = G [(A ∪ B) − (P ∪ Z )];
} Return S.
Factor 3 Approximation Algorithm
I
Theorem-3: Algorithm-1 is a 3-approximation algorithm.
Factor 3 Approximation Algorithm
I
Theorem-3: Algorithm-1 is a 3-approximation algorithm.
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Proof : By induction on number of iterations done by Algorithm-1.
Factor 3 Approximation Algorithm
Algorithm-2 Input: A bipartite graph G = (A ∪ B, E ) and w : A ∪ B → Q+ . Output: A chain deletion set S for G . Compute an optimal solution x for LP(G , w ); Compute S = {v |xv ≥ Return S.
1 2
if v ∈ A} ∪ {v |xv > 0 if v ∈ B};
Second Factor 3 Approximation Algorithm I
The dual of LP(G , w ) is as follows: Pk
j=1 yj
DLP(G , w ) : Maximize Subject to
P
j:v ∈Tj
yj ≤ w (v ), ∀v ∈ V
yj ≥ 0,
1 ≤ j ≤ k.
Second Factor 3 Approximation Algorithm I
The dual of LP(G , w ) is as follows: Pk
j=1 yj
DLP(G , w ) : Maximize Subject to
P
j:v ∈Tj
yj ≤ w (v ), ∀v ∈ V
yj ≥ 0, I
1 ≤ j ≤ k.
αβ-relaxed complementary slackness conditions: Primal: Let α ≥ 1. P For each v ∈ V , either xv = 0 or wα(v ) ≤ j:v ∈Tj yj ≤ w (v ). Dual: Let β ≥ 1. P For each 1 ≤ j ≤ k, either yj = 0 or 1 ≤ v ∈Tj xv ≤ β.
Second Factor 3 Approximation Algorithm I
The dual of LP(G , w ) is as follows: Pk
j=1 yj
DLP(G , w ) : Maximize Subject to
P
j:v ∈Tj
yj ≤ w (v ), ∀v ∈ V
yj ≥ 0, I
1 ≤ j ≤ k.
αβ-relaxed complementary slackness conditions: Primal: Let α ≥ 1. P For each v ∈ V , either xv = 0 or wα(v ) ≤ j:v ∈Tj yj ≤ w (v ). Dual: Let β ≥ 1. P For each 1 ≤ j ≤ k, either yj = 0 or 1 ≤ v ∈Tj xv ≤ β.
I
Theorem-4 [Vazirani 2001] If x and y are primal and dual feasible solutions for LP(G , w ) and DLP(G , w ), respectively, satisfying the αβ-relaxed complementaryPslackness conditions stated as above P then v ∈V w (v )xv ≤ αβ kj=1 yj .
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
I
Proof Let X 0 = χS and y be an optimal solution of DLP(G , w ). We show that x 0 and y satisfy the αβ-complementary slackness conditions with α = 1 and β = 3.
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
I
Proof Let X 0 = χS and y be an optimal solution of DLP(G , w ). We show that x 0 and y satisfy the αβ-complementary slackness conditions with α = 1 and β = 3.
I
Primal complementary slackness condition with α = 1: Since x and y are optimal solutions X ∀v ∈ V , xv > 0 ⇒ yj = w (v ). j:v ∈Tj
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
I
Proof Let X 0 = χS and y be an optimal solution of DLP(G , w ). We show that x 0 and y satisfy the αβ-complementary slackness conditions with α = 1 and β = 3.
I
Primal complementary slackness condition with α = 1: Since x and y are optimal solutions X ∀v ∈ V , xv > 0 ⇒ yj = w (v ). j:v ∈Tj
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Dual Complementary slackness conditions with β = 3 yj > 0 ⇒ |S ∩ Tj | ≤ 3. p,q Let TJ = { u,v 2K2 with |S ∩ Tj | = 4 and yj > 0. ∈A , ∈B } be aP Since yj > 0, by C.S.C. v ∈Tj xv = 1 P Since xu ≥ 12 , xv ≥ 21 , xp > 0, xq > 0, we have v ∈Tj xv > 1.
Factor 2 Approximation Algorithm Algorithm 3 Input: A bipartite graph G = (A ∪ B, E ) and w : A ∪ B → Q+ . Output: A chain deletion set S for G . S = φ; Compute an optimal solution x for LP(G , w ); Compute P(x) = {v |xv ≥ 21 }; while G is not empty and P(x) 6= φ do S = S ∪ P(x); G = G [(A ∪ B) − P(x)]; Remove all the vertices which are not in any 2K2 of G and call this new smaller graph as G ; Compute an optimal solution x for LP(G , w ) and the set P(x);
If G is not an empty graph then compute P(x) = {v |xv > 0 and v ∈ B}; S = S ∪ P(x); Return S.
Factor 2 Approximation Algorithm
I
Lemma-2: The set S returned by Algorithm-3 is a chain deletion set for G .
Factor 2 Approximation Algorithm
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Lemma-2: The set S returned by Algorithm-3 is a chain deletion set for G .
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Proof: If the while loop terminates with an empty graph then clearly S is a chain deletion set. Suppose the graph is not empty at the end of while loop. xv < 21 , for all v ∈ V ⇒ xv > 0, for all v ∈ B. P(x) = {v |xv > 0 and v ∈ B} intersects with all the 2K2 s in the remaining graph G . Hence S is a chain deletion set for the given graph G .
Factor 2 Approximation Algorithm
I
Theorem-6 Algorithm-3 is a factor 2 approximation algorithm.
Factor 2 Approximation Algorithm
I
Theorem-6 Algorithm-3 is a factor 2 approximation algorithm.
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Proof By induction on the number of iterations.
Factor 2 Approximation Algorithm
I
Theorem-6 Algorithm-3 is a factor 2 approximation algorithm.
I
Proof By induction on the number of iterations.
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Base case: w (S) ≤ 2w (S ∗ ), ∀ (G , w ) for which the Algorithm-3 makes exactly one iteration. If P(x) 6=Pφ then S = P(x) P w (S) = v ∈S w (v ) ≤ 2 v ∈S w (v )xv ≤ 2w (S ∗ ). Suppose P(x) = φ. Since G is a bipartite graph 1 ≤ |S ∩ Tj | ≤ 2 for all Tj . χS and y satisfy the ralaxed CSC with α = 1 and β = 2.
Factor 2 Approximation Algorithm
I
Induction Hypothesis: Assume that w (S) ≤ 2w (S ∗ ), ∀(G , w ) for which ALgorithm-3 makes exactly t iterations.
Factor 2 Approximation Algorithm
I
Induction Hypothesis: Assume that w (S) ≤ 2w (S ∗ ), ∀(G , w ) for which ALgorithm-3 makes exactly t iterations.
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Induction Step: We show w (S) ≤ 2w (S ∗ ), ∀(G , w ) for which the Algorithm-3 makes exactly t + 1 iterations. 6 φ (as the algo makes more than one iteration). P1 = {v |xv ≥ 21 } = G 0 = G [V − (P1 ∪ Z1 )] and w 0 is the input to the algorithm for the 2nd iteration. Now the algorithm makes exactly t iteration for the input (G 0 , w 0 ). 0 By induction hypothesis we have w (S 0 ) ≤ 2w (S ∗ ).
Factor 2 Approximation Algorithm I
w (S ∗ ) ≥
X
w (v )xv
=
v ∈V
X v ∈P1
≥
X
w (v )xv +
1 w (P1 ) + 2
w (v )xv
v ∈V −P1
X v ∈V −P1
w (v )xv
Factor 2 Approximation Algorithm I
w (S ∗ ) ≥
X
w (v )xv
=
v ∈V
X v ∈P1
≥
X
w (v )xv +
1 w (P1 ) + 2
w (v )xv
v ∈V −P1
X
w (v )xv
v ∈V −P1
I
X
w (v )xv
≥
v ∈V −P1
w (S ∗ ) ≥ Hence w (S) ≤ 2w (S ∗ ).
1 w (S 0 ) 2 1 1 1 w (P1 ) + w (S 0 ) = w (S) 2 2 2
Maximum Induced Chain Subgraph
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Maximum Induced Chain Subgraph: Given a bipartite graph + G = (V , E ) and a weight function P w : V → Q , find a set S ⊆ V of maximum weight w (S) = v ∈S w (v ) such that G [S] is a chain graph.
Maximum Induced Chain Subgraph
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Maximum Induced Chain Subgraph: Given a bipartite graph + G = (V , E ) and a weight function P w : V → Q , find a set S ⊆ V of maximum weight w (S) = v ∈S w (v ) such that G [S] is a chain graph.
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Maximum Induced Chain Subgraph can be approximated within a factor of 1.5.
Maximum Induced Chain Subgraph
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Maximum Induced Chain Subgraph: Given a bipartite graph + G = (V , E ) and a weight function P w : V → Q , find a set S ⊆ V of maximum weight w (S) = v ∈S w (v ) such that G [S] is a chain graph.
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Maximum Induced Chain Subgraph can be approximated within a factor of 1.5.
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Minimum Chain Vertex Deletion and Maximum Induced Chain Subgraph are APX-complete.
THANK YOU
Sounaka Mishra
1
Mrinal Kumar1 Joint Work With N. Safina Devi 1
Saket Saurabh2
Indian Institute of Technology Madras, Chennai, India.1 The Institute of Mathematical Sciences, Chennai, India.2
19th February, 2011
Chain Graph I
Chain Graph: A bipartite graph G = (A ∪ B, E ) is called a chain graph if there exists an ordering ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } such that N(v1 ) ⊆ N(v2 ) . . . ⊆ N(vn ).
Chain Graph I
Chain Graph: A bipartite graph G = (A ∪ B, E ) is called a chain graph if there exists an ordering ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } such that N(v1 ) ⊆ N(v2 ) . . . ⊆ N(vn ).
I
Example: a
x
b
y
c
z
N (c) ⊆ N (b) ⊆ N (a) N (x) ⊆ N (y) ⊆ N (z)
Chain Graph I
Chain Graph: A bipartite graph G = (A ∪ B, E ) is called a chain graph if there exists an ordering ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } such that N(v1 ) ⊆ N(v2 ) . . . ⊆ N(vn ).
I
Example: a
x
b
y
c
z
N (c) ⊆ N (b) ⊆ N (a) N (x) ⊆ N (y) ⊆ N (z)
I
Observation : If G = (A ∪ B, E ) is a chain graph and ρ =< v1 , v2 , . . . , vn > of the vertices in A = {v1 , . . . , vn } then vertices in B has a similar nested ordering.
Characterizations of Chain Graph
I
Theorem-1: [Yannakakis 1981] For a bipartite graph G = (A ∪ B, E ) the following are equivalent (a) G is a chain graph, (b) G is 2K2 -free, (c) C (G ) is chordal, (d) The bipartite complement of G is a chain graph.
Characterizations of Chain Graph
I
Theorem-1: [Yannakakis 1981] For a bipartite graph G = (A ∪ B, E ) the following are equivalent (a) G is a chain graph, (b) G is 2K2 -free, (c) C (G ) is chordal, (d) The bipartite complement of G is a chain graph.
I
Corollary-1: Given a bipartite graph G , it can be tested in polynomial time whether G is chain.
Characterizations of Chain Graph
I
Theorem-1: [Yannakakis 1981] For a bipartite graph G = (A ∪ B, E ) the following are equivalent (a) G is a chain graph, (b) G is 2K2 -free, (c) C (G ) is chordal, (d) The bipartite complement of G is a chain graph.
I
Corollary-1: Given a bipartite graph G , it can be tested in polynomial time whether G is chain.
I
For all pair of edges (u, v ) and (p, q) in G , test whether G [{u, v , p, q}] is isomerphic to 2K2 .
Minimum Chain vertex Deletion
I
Minimum Chain Vertex Deletion: Given a bipartite graph G = (V , E ) and a weight function w : V → Q+ , find a set S ⊆ V P of minimum weight w (S) = v ∈S w (v ) such that G [V \ S] is a chain graph.
Minimum Chain vertex Deletion
I
Minimum Chain Vertex Deletion: Given a bipartite graph G = (V , E ) and a weight function w : V → Q+ , find a set S ⊆ V P of minimum weight w (S) = v ∈S w (v ) such that G [V \ S] is a chain graph.
I
Theorem-2 [Yannakakis 1981] Minimum Chain Vertex Deletion is NP-complete.
Minimum Chain vertex Deletion
I
Minimum Chain Vertex Deletion: Given a bipartite graph G = (V , E ) and a weight function w : V → Q+ , find a set S ⊆ V P of minimum weight w (S) = v ∈S w (v ) such that G [V \ S] is a chain graph.
I
Theorem-2 [Yannakakis 1981] Minimum Chain Vertex Deletion is NP-complete.
I
Minimum Chain Vertex Deletion can be approximated within a factor 4.
Minimum Chain vertex Deletion I
ILP formulation of Minimum Chain Vertex Deletion Given a graph G = (V , E ), let {T1 , T2 , . . . Tk } be the set of all 2K2 s in G . We assume that ∀ v ∈ V there is a Ti containing v . P ILP(G , w ) : Minimize w (v )xv P v ∈V Subject to v ∈Tj xv ≥ 1, 1 ≤ j ≤ k xv ∈ {0, 1},
∀v ∈ V .
Minimum Chain vertex Deletion I
ILP formulation of Minimum Chain Vertex Deletion Given a graph G = (V , E ), let {T1 , T2 , . . . Tk } be the set of all 2K2 s in G . We assume that ∀ v ∈ V there is a Ti containing v . P ILP(G , w ) : Minimize w (v )xv P v ∈V Subject to v ∈Tj xv ≥ 1, 1 ≤ j ≤ k xv ∈ {0, 1},
I
∀v ∈ V .
LP relaxation of ILP LP(G , w ) : Minimize Subject to
P
P v ∈V
v ∈Tj
w (v )xv xv ≥ 1, 1 ≤ j ≤ k
xv ≥ 0,
∀v ∈ V .
Minimum Chain vertex Deletion
I
Lemma-1: Let x be an optimal solution of LP(G , w ) for a given bipartite graph G = (A ∪ B, E ). If xv < 13 , for all v ∈ A ∪ B, then one can compute another optimal solution x 0 for LP(G , w ) with x 0 (v ) ≥ 31 , for some v ∈ A ∪ B.
Factor 3 Approximation Algorithm Algorithm 1 Input: A bipartite graph G = (A ∪ B, E ) and w : A ∪ B → Q+ . Output: A chain deletion set S for G . S = φ; while G has at least one 2K2 do { Construct LP(G , w ) and compute an optimal solution x satisfying the property in Lemma-1; compute the sets P = {v |xv ≥ 31 }, Z = {v |xv = 0}; S = S ∪ P; G = G [(A ∪ B) − (P ∪ Z )];
} Return S.
Factor 3 Approximation Algorithm
I
Theorem-3: Algorithm-1 is a 3-approximation algorithm.
Factor 3 Approximation Algorithm
I
Theorem-3: Algorithm-1 is a 3-approximation algorithm.
I
Proof : By induction on number of iterations done by Algorithm-1.
Factor 3 Approximation Algorithm
Algorithm-2 Input: A bipartite graph G = (A ∪ B, E ) and w : A ∪ B → Q+ . Output: A chain deletion set S for G . Compute an optimal solution x for LP(G , w ); Compute S = {v |xv ≥ Return S.
1 2
if v ∈ A} ∪ {v |xv > 0 if v ∈ B};
Second Factor 3 Approximation Algorithm I
The dual of LP(G , w ) is as follows: Pk
j=1 yj
DLP(G , w ) : Maximize Subject to
P
j:v ∈Tj
yj ≤ w (v ), ∀v ∈ V
yj ≥ 0,
1 ≤ j ≤ k.
Second Factor 3 Approximation Algorithm I
The dual of LP(G , w ) is as follows: Pk
j=1 yj
DLP(G , w ) : Maximize Subject to
P
j:v ∈Tj
yj ≤ w (v ), ∀v ∈ V
yj ≥ 0, I
1 ≤ j ≤ k.
αβ-relaxed complementary slackness conditions: Primal: Let α ≥ 1. P For each v ∈ V , either xv = 0 or wα(v ) ≤ j:v ∈Tj yj ≤ w (v ). Dual: Let β ≥ 1. P For each 1 ≤ j ≤ k, either yj = 0 or 1 ≤ v ∈Tj xv ≤ β.
Second Factor 3 Approximation Algorithm I
The dual of LP(G , w ) is as follows: Pk
j=1 yj
DLP(G , w ) : Maximize Subject to
P
j:v ∈Tj
yj ≤ w (v ), ∀v ∈ V
yj ≥ 0, I
1 ≤ j ≤ k.
αβ-relaxed complementary slackness conditions: Primal: Let α ≥ 1. P For each v ∈ V , either xv = 0 or wα(v ) ≤ j:v ∈Tj yj ≤ w (v ). Dual: Let β ≥ 1. P For each 1 ≤ j ≤ k, either yj = 0 or 1 ≤ v ∈Tj xv ≤ β.
I
Theorem-4 [Vazirani 2001] If x and y are primal and dual feasible solutions for LP(G , w ) and DLP(G , w ), respectively, satisfying the αβ-relaxed complementaryPslackness conditions stated as above P then v ∈V w (v )xv ≤ αβ kj=1 yj .
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
I
Proof Let X 0 = χS and y be an optimal solution of DLP(G , w ). We show that x 0 and y satisfy the αβ-complementary slackness conditions with α = 1 and β = 3.
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
I
Proof Let X 0 = χS and y be an optimal solution of DLP(G , w ). We show that x 0 and y satisfy the αβ-complementary slackness conditions with α = 1 and β = 3.
I
Primal complementary slackness condition with α = 1: Since x and y are optimal solutions X ∀v ∈ V , xv > 0 ⇒ yj = w (v ). j:v ∈Tj
Second Factor 3 Approximation Algorithm I
Theorem-5: Algorithm-2 is a 3-approximation algorithm.
I
Proof Let X 0 = χS and y be an optimal solution of DLP(G , w ). We show that x 0 and y satisfy the αβ-complementary slackness conditions with α = 1 and β = 3.
I
Primal complementary slackness condition with α = 1: Since x and y are optimal solutions X ∀v ∈ V , xv > 0 ⇒ yj = w (v ). j:v ∈Tj
I
Dual Complementary slackness conditions with β = 3 yj > 0 ⇒ |S ∩ Tj | ≤ 3. p,q Let TJ = { u,v 2K2 with |S ∩ Tj | = 4 and yj > 0. ∈A , ∈B } be aP Since yj > 0, by C.S.C. v ∈Tj xv = 1 P Since xu ≥ 12 , xv ≥ 21 , xp > 0, xq > 0, we have v ∈Tj xv > 1.
Factor 2 Approximation Algorithm Algorithm 3 Input: A bipartite graph G = (A ∪ B, E ) and w : A ∪ B → Q+ . Output: A chain deletion set S for G . S = φ; Compute an optimal solution x for LP(G , w ); Compute P(x) = {v |xv ≥ 21 }; while G is not empty and P(x) 6= φ do S = S ∪ P(x); G = G [(A ∪ B) − P(x)]; Remove all the vertices which are not in any 2K2 of G and call this new smaller graph as G ; Compute an optimal solution x for LP(G , w ) and the set P(x);
If G is not an empty graph then compute P(x) = {v |xv > 0 and v ∈ B}; S = S ∪ P(x); Return S.
Factor 2 Approximation Algorithm
I
Lemma-2: The set S returned by Algorithm-3 is a chain deletion set for G .
Factor 2 Approximation Algorithm
I
Lemma-2: The set S returned by Algorithm-3 is a chain deletion set for G .
I
Proof: If the while loop terminates with an empty graph then clearly S is a chain deletion set. Suppose the graph is not empty at the end of while loop. xv < 21 , for all v ∈ V ⇒ xv > 0, for all v ∈ B. P(x) = {v |xv > 0 and v ∈ B} intersects with all the 2K2 s in the remaining graph G . Hence S is a chain deletion set for the given graph G .
Factor 2 Approximation Algorithm
I
Theorem-6 Algorithm-3 is a factor 2 approximation algorithm.
Factor 2 Approximation Algorithm
I
Theorem-6 Algorithm-3 is a factor 2 approximation algorithm.
I
Proof By induction on the number of iterations.
Factor 2 Approximation Algorithm
I
Theorem-6 Algorithm-3 is a factor 2 approximation algorithm.
I
Proof By induction on the number of iterations.
I
Base case: w (S) ≤ 2w (S ∗ ), ∀ (G , w ) for which the Algorithm-3 makes exactly one iteration. If P(x) 6=Pφ then S = P(x) P w (S) = v ∈S w (v ) ≤ 2 v ∈S w (v )xv ≤ 2w (S ∗ ). Suppose P(x) = φ. Since G is a bipartite graph 1 ≤ |S ∩ Tj | ≤ 2 for all Tj . χS and y satisfy the ralaxed CSC with α = 1 and β = 2.
Factor 2 Approximation Algorithm
I
Induction Hypothesis: Assume that w (S) ≤ 2w (S ∗ ), ∀(G , w ) for which ALgorithm-3 makes exactly t iterations.
Factor 2 Approximation Algorithm
I
Induction Hypothesis: Assume that w (S) ≤ 2w (S ∗ ), ∀(G , w ) for which ALgorithm-3 makes exactly t iterations.
I
Induction Step: We show w (S) ≤ 2w (S ∗ ), ∀(G , w ) for which the Algorithm-3 makes exactly t + 1 iterations. 6 φ (as the algo makes more than one iteration). P1 = {v |xv ≥ 21 } = G 0 = G [V − (P1 ∪ Z1 )] and w 0 is the input to the algorithm for the 2nd iteration. Now the algorithm makes exactly t iteration for the input (G 0 , w 0 ). 0 By induction hypothesis we have w (S 0 ) ≤ 2w (S ∗ ).
Factor 2 Approximation Algorithm I
w (S ∗ ) ≥
X
w (v )xv
=
v ∈V
X v ∈P1
≥
X
w (v )xv +
1 w (P1 ) + 2
w (v )xv
v ∈V −P1
X v ∈V −P1
w (v )xv
Factor 2 Approximation Algorithm I
w (S ∗ ) ≥
X
w (v )xv
=
v ∈V
X v ∈P1
≥
X
w (v )xv +
1 w (P1 ) + 2
w (v )xv
v ∈V −P1
X
w (v )xv
v ∈V −P1
I
X
w (v )xv
≥
v ∈V −P1
w (S ∗ ) ≥ Hence w (S) ≤ 2w (S ∗ ).
1 w (S 0 ) 2 1 1 1 w (P1 ) + w (S 0 ) = w (S) 2 2 2
Maximum Induced Chain Subgraph
I
Maximum Induced Chain Subgraph: Given a bipartite graph + G = (V , E ) and a weight function P w : V → Q , find a set S ⊆ V of maximum weight w (S) = v ∈S w (v ) such that G [S] is a chain graph.
Maximum Induced Chain Subgraph
I
Maximum Induced Chain Subgraph: Given a bipartite graph + G = (V , E ) and a weight function P w : V → Q , find a set S ⊆ V of maximum weight w (S) = v ∈S w (v ) such that G [S] is a chain graph.
I
Maximum Induced Chain Subgraph can be approximated within a factor of 1.5.
Maximum Induced Chain Subgraph
I
Maximum Induced Chain Subgraph: Given a bipartite graph + G = (V , E ) and a weight function P w : V → Q , find a set S ⊆ V of maximum weight w (S) = v ∈S w (v ) such that G [S] is a chain graph.
I
Maximum Induced Chain Subgraph can be approximated within a factor of 1.5.
I
Minimum Chain Vertex Deletion and Maximum Induced Chain Subgraph are APX-complete.
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