On the Optimality of a Class of LP-based Algorithms

In this paper we will be concerned with a class of packing and covering problems which includes Vertex Cover and Independent Set. Typically, one can write an LP relaxation and then round the solution. For instance, for Vertex Cover one can obtain a 2-approximation via this approach. On the other hand, Khot and Regev [KR08] proved that, assuming the Unique Games Conjecture (UGC), it is NP-hard to approximate Vertex Cover to within a factor better than 2 − ε for any constant ε > 0. From their, and subsequent proofs of this result, it was not clear why this LP relaxation should be optimal. The situation was akin to Maximum Cut, where a natural SDP relaxation for it was proved by Khot et al. [KKMO07] to be optimal assuming the UGC. A beautiful result of Raghavendra [Rag08] explains why the SDP is optimal (assuming the UGC). Moreover, his result generalizes to a large class of constraint satisfaction problems (CSPs). Unfortunately, we do not know how to extend his framework so that it applies for problems such as Vertex Cover where the constraints are strict. In this paper, we explain why the simple LP-based rounding algorithm for the Vertex Cover problem is optimal assuming the UGC. Complementing Raghavendra’s result, our result generalizes to a class of strict, covering/packing type CSPs. We first write down a natural LP relaxation for this class of problems and present a simple rounding algorithm for it. The key ingredient, then, is a dictatorship test, which is parametrized by a rounding-gap example for this LP, whose completeness and soundness are the LP-value and the rounded value respectively. To the best of our knowledge, ours is the first result which proves the optimality of LP-based rounding algorithms systematically. This work was done while the authors were at Microsoft Research India, Bangalore Dept. of Computer Science and Engineering, IIT Delhi, New Delhi, email: [email protected] Department of Computer Science, Princeton University, email: [email protected]. Supported by NSF grants 0830673, 0832797, 528414 Institute for Advanced Study, Princeton, email: [email protected]. Supported by NSF grant 0832797 and IAS sub-contract no. 00001583. Microsoft Research India, Bangalore, email: [email protected]