Exponential lower bounds and integrality gaps for tree-like Lovász-Schrijver procedures

The matrix cuts of Lovász and Schrijver are methods for tightening linear relaxations of zero-one programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems with strong guarantees, and to solve certain instances of Boolean satisfiability. We show that relaxations of linear programs, obtained by tightening via any subexponential-size semidefinite Lovász-Schrijver derivation tree, cannot approximate max-k-SAT to a factor better than 1 + 1 2k−1 , max-k-XOR to a factor better than 2− ε, nor vertex cover to a factor better than 7/6. We prove exponential size lower bounds for tree-like Lovász-Schrijver proofs of unsatisfiability for several prominent unsatisfiable CNFs, including random 3-CNF formulas, random systems of linear equations, and the Tseitin graph formulas. Furthermore, we prove that tree-like LS+ cannot polynomially simulate tree-like cutting planes, and that tree-like LS+ cannot polynomially simulate unrestricted resolution. All of our size lower bounds for derivation trees are based upon connections between the size and height of the derivation tree (its rank). The primary method is a tree-size/rank trade-off for LovászSchrijver refutations: Small tree size implies small rank. Surprisingly, this does not hold for derivations of arbitrary linear inequalities. We show that for LS0 and LS, there are examples with polynomial-size tree-like derivations, but requiring linear rank.