Summary: Towards strong nonapproximability results in the Lov’aszSchrijver
Mikhail Alekhnovich # Sanjeev Arora + Iannis Tourlakis #
Lov’asz and Schrijver described a generic method of tightening the LP and SDP relaxation for
any 01 optimization problem. These tightened relaxations were the basis of several celebrated
approximation algorithms (such as for MAXCUT, MAX3SAT, and SPARSEST CUT).
We prove strong nonapproximability results in this model for wellknown problems such as
MAX3SAT, Hypergraph Vertex Cover and Minimum Set Cover. We show that the relaxations
produced by as many as ## n) rounds of the LS+ procedure do not allow nontrivial approxima
tion, thus ruling out the possibility that the LS+ approach gives even slightly subexponential
approximation algorithms for these problems.
We also point out why our results are somewhat incomparable to known nonapproximability
results proved using PCPs, and formalize several interesting open questions.
The past decade has seen a dramatic improvement of our understanding of the approximation
properties of many NPhard optimization problems. Many new approximation algorithms were
designed, especially using linear programming (LP) or semidefinite programming (SDP) relax
ations. For many problems it was proved using probabilistically checkable proofs (PCPs) that
these algorithms are the best possible polynomialtime algorithms unless NP has subexponential