Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász-Schrijver Procedures

• Authors: Toniann Pitassi;Nathan Segerlind
• Venue: SIAM Journal on Computing
• Year: 2012

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, and we solve certain instances of Boolean satisfiability. Our first result is a size/rank tradeoff for tree-like Lovász-Schrijver refutations, showing that any refutation that has small size also has small rank. This allows us to immediately derive exponential-size lower bounds for tree-like refutations of many unsatisfiable systems of inequalities where, prior to our work, only strong rank bounds were known. Unfortunately, we show that this tradeoff does not hold more generally for derivations of arbitrary inequalities. We give a very simple example showing that derivations can be very small but nonetheless require maximal rank. This rules out a generic argument for obtaining a size-based integrality gap from the corresponding rank-based integrality gap. Our second contribution is to show that a modified argument can often be used to prove size-based integrality gaps from rank-based integrality gaps. We apply this method to prove size-based integrality gaps for several prominent examples where, prior to our work, only rank-based integrality gaps were known. Our third contribution is to prove new separation results. Using our machinery for converting rank-based lower bounds and integrality gaps into size-based lower bounds, we show that tree-like $\mbox{LS}_+$ cannot polynomially simulate tree-like cutting planes, and that tree-like $\mbox{LS}_+$ cannot polynomially simulate resolution.