New Lower Bounds for Vertex Cover in the Lovasz-Schrijver Hierarchy

Lovdsz and Schrijver (1991) defined three progressively stronger procedures LS0, LS and LS+, for systematically tightening linear relaxations over many rounds. All three procedures yield the integral hull after at most n rounds. On the other hand, constant rounds of LS+ can derive the relaxations behind many famous approximation algorithms such as those for MAX-CUT, SPARSEST-CUT. So proving round lower bounds for these procedures on specific problems may give evidence about inapproximability. We prove new round lower bounds for vertex cover in the LS hierarchy. Arora et al. (2006) showed that the integrality gap for VERTEX COVER relaxations remains 2 - o(1) even after Omega(log n) rounds LS. However, their method can only prove round lower bounds as large as the girth of the input graph, which is O(log n) for interesting graphs. We break through this "girth barrier" and show that the integrality gap for VERTEX COVER remains 1.5 - epsi even after Omega(log2 n) rounds of LS. In contrast, the best PCP-based results only rule out 1.36-approximations. Moreover, we conjecture that the new technique we introduce to prove our lower bound, the "fence" method, may lead to linear or nearly linear LS round lower bounds for VERTEX COVER

[1]  Sanjeev Arora,et al.  Towards Strong Nonapproximability Results in the Lovász-Schrijver Hierarchy , 2005, STOC '05.

[2]  Iannis Tourlakis.  Towards Optimal Integrality Gaps for Hypergraph Vertex Cover in the Lovász-Schrijver Hierarchy , 2005, APPROX-RANDOM.

[3]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[4]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[5]  Toniann Pitassi,et al.  Rank Bounds and Integrality Gaps for Cutting Planes Procedures , 2006, Theory Comput..

[6]  Robert Krauthgamer,et al.  The Probable Value of the Lovász--Schrijver Relaxations for Maximum Independent Set , 2003, SIAM J. Comput..

[7]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[8]  S. Safra,et al.  On the hardness of approximating minimum vertex cover , 2005 .

[9]  William J. Cook,et al.  On the Matrix-Cut Rank of Polyhedra , 2001, Math. Oper. Res..

[10]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[11]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[12]  Tamon Stephen,et al.  On a Representation of the Matching Polytope Via Semidefinite Liftings , 1999, Math. Oper. Res..

[13]  Michel X. Goemans,et al.  When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures? , 2001, Math. Oper. Res..

[14]  Subhash Khot.  On the power of unique 2-prover 1-round games , 2002, STOC '02.

[15]  Béla Bollobás,et al.  Proving Integrality Gaps without Knowing the Linear Program , 2006, Theory Comput..

[16]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

[17]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..