Critical graphs and facets of the stable set polytope László Lipták The Fields Institute University of Toronto Toronto, Ontario, Canada lliptak@fields.utoronto.ca László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com

Independent (stable) sets maximum size of an independent set of nodes Difficult (NP-hard)!

α-critical graphs [no isolated nodes; connected]

Erdős - Gallai Hajnal : defect of G Gallai odd cycles Hajnal even subdivisions of K4 Andrásfai

criticality is preserved by even subdivision d is preserved enough to find graphs w/o

+ = + = + = Composing α–critical graphs Gallai, Plummer, Wessel basis graphs Enough to find graphs with deg >2

Andrásfai Surányi, L there is a finite number of basis graphs L

The stable set polytope c a b c b G a

Facets of the stable set polytope G b a b a c

(1) and (2) suffice iff G is bipartite (nonnegativity constraints) (edge constraints) (1) and (2) suffice iff G is bipartite b b a c a

 (clique constraints) (1) and (3) suffice iff G is perfect Fulkerson-Chvátal (odd hole constraints) (rank constraints) G -critical  rank constraint for U=V is a facet View facets as generalizations of -critical graphs Chvátal

nontrivial facet of STAB(G): vertex of this facet: stable set integrality gap of this facet: defect of facet non-trivial L-Schrijver

critical facet of STAB(G): deleting any edge, does not remain valid 1 2 2 1 1 2 1 1 a a a a irreducible facet of STAB(G): no

In every critical facet aid Structural significance of d In every critical facet aid Sewell In every critical facet: ● deg(i)  ai+d  2 d ● for d>1, deg(i)  2 d+1 LL-LL

Finite basis theorem for facets For d=2, all critical facets are subdivisions of K4 Sewell 2 2 2 2 2 2 2 2 2 2 For every d, there is a finite number of irreducible critical facets with defect d. LL-LL

Deriving facets trivial constraints (edge constraints)

1 5 2 4 3 (odd hole constraints)

1 5 2 4 3 (clique constraints)

Every facet can be obtained by iterating this process at most n times L-Schrijver Even without squaring Linear objective functions subject to all constraints derived in c iterations can be optimized in polynomial time L-Schrijver

rank of facet: smallest c Which facets can be derived in c iterations? rank of facet: smallest c r: rank of d : defect of L-Schrijver

The average degree of a critical facet odd hole contraints ? The average degree of a critical facet with rank r is <2r. Lipták