Proving Integrality Gaps without Knowing the Linear Program

Abstract

During the past decade we have had much success in proving (using probabilistically checkable proofs or PCPs) that computing approximate solutions to NP-hard optimization problems such as CLIQUE, COLORING, SET-COVER etc. is no easier than computing optimal solutions.

After the above notable successes, this effort is now stuck for many other problems, such as METRIC TSP, VERTEX COVER, GRAPH EXPANSION, etc.

In a recent paper with Béla Bollobás and László Lovász we argue that NP-hardness of approximation may be too ambitious a goal in these cases, since NP-hardness implies a lowerbound – assuming P ≠ NP – on all polynomial time algorithms. A less ambitious goal might be to prove a lowerbound on restricted families of algorithms. Linear and semidefinite programs constitute a natural family, since they are used to design most approximation algorithms in practice. A lowerbound result for a large subfamily of linear programs may then be viewed as a lowerbound for a restricted computational model, analogous say to lowerbounds for monotone circuits

The above paper showed that three fairly general families of linear relaxations for vertex cover cannot be used to design a 2-approximation for Vertex Cover. Our methods seem relevant to other problems as well.

This talk surveys this work, as well as other open problems in the field. The most interesting families of relaxations involve those obtained by the so-called lift and project methods of Lovász-Schrijver and Sherali-Adams.

Proving lowerbounds for such linear relaxations involve elements of combinatorics (i.e., strong forms of classical Erdős theorems), proof complexity, and the theory of convex sets.