Extended Formulations of Stable Set Polytopes via Decomposition Michele Conforti (U Padova), Bert Gerards (CIW Amsterdam), Kanstantsin Pashkovich (U Padova) Aussois, January 2013 1/ 30
History Gr¨ otschel, Lov´ asz, Schrijver 1981 A polynomial time algorithm that computes a stable set of maximum weight in a perfect graph based on the ellipsoid method. Gr¨ otschel, Lov´ asz, Schrijver 1986 A compact SDP-extended formulation for the stable set polytope of perfect graphs. Chudnovsky, Robertson, Seymour, Thomas 2003 The Strong Perfect Graph Theorem. 2/ 30
Outline Introduction 1 Clique Cutset Decomposition 2 Amalgam Decomposition 3 Template Decomposition 4 Applying Decompositions for Cap-Free Odd-Signable Graphs 5 3/ 30
Stable Set Polytope Stable Set Polytope The stable set polytope P stable ( G ) ⊆ R E of the graph G = ( V , E ) is defined by P stable ( G ) = conv( { χ ( S ) : S is a stable set in G } ) . 4/ 30
Stable Set Polytope Stable Set Polytope The stable set polytope P stable ( G ) ⊆ R E of the graph G = ( V , E ) is defined by P stable ( G ) = conv( { χ ( S ) : S is a stable set in G } ) . 4/ 30
Extended Formulations of Polytopes Extension A polyhedron Q ⊆ R d and a linear projection p : R d → R m form an extension of a polytope P ⊆ R m if P = p ( Q ) holds. the size of the extension is the number of facets of Q . Crucial Fact For each c ∈ R m , we have max {� c , x � : x ∈ P } = max {� T t c , y � : y ∈ Q } if the linear map p : R d → R m is defined as p ( y ) = Ty . 5/ 30
Decomposition Decomposition A decomposition of an object X is the substitution of X with objects, according to a given decomposition rule R . These objects are the blocks of the decomposition of X with R . 6/ 30
Constructing Extended Formulations via Decomposition Class of Objects Given a rule R , a class of objects C and a class of objects P , we say that C is decomposable into P with R if every object in C can be recursively decomposed with R until all blocks belong to P . 7/ 30
Constructing Extended Formulations via Decomposition Class of Objects Given a rule R , a class of objects C and a class of objects P , we say that C is decomposable into P with R if every object in C can be recursively decomposed with R until all blocks belong to P . Extended Formulations via Decomposition For every object X in C there exists a compact extended formulation of the polytope P ( X ) if For every object Y in C which is decomposed by the rule R into objects Y 1 , Y 2 ,. . . , Y k with extended formulations for P ( Y 1 ), P ( Y 2 ),. . . , P ( Y k ) of size s 1 , s 2 ,. . . , s k there is an extended formulation for the polytope P ( Y ) of size s 1 + s 2 + . . . + s k . 7/ 30
Constructing Extended Formulations via Decomposition Class of Objects Given a rule R , a class of objects C and a class of objects P , we say that C is decomposable into P with R if every object in C can be recursively decomposed with R until all blocks belong to P . Extended Formulations via Decomposition For every object X in C there exists a compact extended formulation of the polytope P ( X ) if For every object Y in C which is decomposed by the rule R into objects Y 1 , Y 2 ,. . . , Y k with extended formulations for P ( Y 1 ), P ( Y 2 ),. . . , P ( Y k ) of size s 1 , s 2 ,. . . , s k there is an extended formulation for the polytope P ( Y ) of size s 1 + s 2 + . . . + s k . There is a recursive decomposition of every object in C by the rule R results into polynomial number of objects in P . 7/ 30
Constructing Extended Formulations via Decomposition Class of Objects Given a rule R , a class of objects C and a class of objects P , we say that C is decomposable into P with R if every object in C can be recursively decomposed with R until all blocks belong to P . Extended Formulations via Decomposition For every object X in C there exists a compact extended formulation of the polytope P ( X ) if For every object Y in C which is decomposed by the rule R into objects Y 1 , Y 2 ,. . . , Y k with extended formulations for P ( Y 1 ), P ( Y 2 ),. . . , P ( Y k ) of size s 1 , s 2 ,. . . , s k there is an extended formulation for the polytope P ( Y ) of size s 1 + s 2 + . . . + s k . There is a recursive decomposition of every object in C by the rule R results into polynomial number of objects in P . For every object Y in P there exists a compact extended formulation of the polytope P ( Y ). 7/ 30
Outline Introduction 1 Clique Cutset Decomposition 2 Amalgam Decomposition 3 Template Decomposition 4 Applying Decompositions for Cap-Free Odd-Signable Graphs 5 8/ 30
Clique Cutset Cutset A clique K ⊆ V of G = ( V , E ) is a clique cutset if V \ K can be partitioned into two nonempty sets V 1 and V 2 such that no node of V 1 is adjacent to V 2 . Clique Cutset Decomposition The blocks of the clique cutset decomposition are the subgraphs G 1 and G 2 of G induced by V 1 ∪ K and V 2 ∪ K , respectively. 9/ 30
Clique Cutset Decomposition G G 1 G 2 10/ 30
Stable Set Polytope and Clique Cutset Chv´ atal 1975 A point lies in P stable ( G ) if only if its restriction to V 1 ∪ K lies in P stable ( G 1 ) and its restriction to V 2 ∪ K lies in P stable ( G 2 ). 11/ 30
Stable Set Polytope and Clique Cutset Chv´ atal 1975 A point lies in P stable ( G ) if only if its restriction to V 1 ∪ K lies in P stable ( G 1 ) and its restriction to V 2 ∪ K lies in P stable ( G 2 ). Proof Let x be a point such that its restriction x 1 to V 1 ∪ K lies in P stable ( G 1 ) and its restriction x 2 to V 2 ∪ K lies in P stable ( G 2 ). Then, x i = � λ i S χ ( S ) S ∈S ( G i ) where λ i ≥ 0, � S ∈S ( G i ) λ i S = 1. Thus, for every v ∈ K � λ 1 � λ 2 S = S . S ∈S ( G 1 ) S ∈S ( G 2 ) v ∈ S v ∈ S 11/ 30
Clique Cutset Decompositions Clique Cutset Decomposition If G has a clique cutset then there exists a clique cutset decomposition of G such that one of the blocks does not have a clique cutset. G G 1 G 2 12/ 30
Outline Introduction 1 Clique Cutset Decomposition 2 Amalgam Decomposition 3 Template Decomposition 4 Applying Decompositions for Cap-Free Odd-Signable Graphs 5 13/ 30
Amalgam Amalgam Triple ( A , K , B ) is an amalgam of a graph G = ( V , E ) if V can be partitioned into V 1 , V 2 and K such that | V 1 | ≥ 2 and | V 2 | ≥ 2 and K is a (possibly empty) clique. V 1 and V 2 contain nonempty subsets A and B such that: K is universal to A and B A and B are universal V 1 \ A and V 2 \ B are nonadjacent. K V 2 V 1 B A 14/ 30
Amalgam Decomposition Blocks of Amalgam Decomposition The blocks of the amalgam decomposition of G with ( A , K , B ) are the graph obtained by adding a new node b to the subgraph of G induced by V 1 ∪ K and adding edges from b to each of the nodes in K ∪ A the graph obtained by adding a new node a to the subgraph of G induced by V 2 ∪ K and adding edges from a to each of the nodes in K ∪ B . K V 1 A b 15/ 30
Amalgam Decomposition Blocks of Amalgam Decomposition The blocks of the amalgam decomposition of G with ( A , K , B ) are the graph obtained by adding a new node b to the subgraph of G induced by V 1 ∪ K and adding edges from b to each of the nodes in K ∪ A the graph obtained by adding a new node a to the subgraph of G induced by V 2 ∪ K and adding edges from a to each of the nodes in K ∪ B . K V 2 B a 15/ 30
Extended Formulation via Amalgam Decomposition K V 2 V 1 B A a b K V 2 V 1 B A 16/ 30
Extended Formulation via Amalgam Decomposition Conforti, Gerards, P. 2012 A point lies in P stable ( G ) if and only if it can be extended by x a , x b such that its restriction to V 1 ∪ K ∪ { a } lies in P stable ( G 1 ) and its restriction to V 2 ∪ K ∪ { b } lies in P stable ( G 2 ) and x a + x b + � v ∈ K x v = 1. K V 2 V 1 B A a b 17/ 30
Extended Formulation via Amalgam Decomposition K V 1 A a b K V 2 B a 17/ 30
Extended Formulation via Amalgam Decomposition K K V 1 A a b b K V 2 B a 17/ 30
Amalgam Decomposition 18/ 30
Amalgam Decomposition a 1 u b a 2 Conforti, Gerards, P. 2012 Every recursive amalgam decomposition results into polynomial number of graphs without an amalgam and polynomial number of cliques (no clique is decomposed during the recursion). 18/ 30
Outline Introduction 1 Clique Cutset Decomposition 2 Amalgam Decomposition 3 Template Decomposition 4 Applying Decompositions for Cap-Free Odd-Signable Graphs 5 19/ 30
Cutset Decomposition Cutset A vertex set K ⊆ V of G = ( V , E ) is a cutset if V \ K can be partitioned into two nonempty sets V 1 and V 2 such that no node of V 1 is adjacent to V 2 . Cutset Decomposition The blocks of the cutset decomposition are the subgraphs G 1 and G 2 of G induced by V 1 ∪ K and V 2 ∪ K , respectively. 20/ 30
Stable Set Polytope and Cutset G G 1 G 2 21/ 30
Stable Set Polytope and Cutset G G 1 G 2 21/ 30
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