Improved Approximation Algorithms for Projection Games

The projection games (aka Label-Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label-Cover. In this paper we design several approximation algorithms for projection games: 1. A polynomial-time approximation algorithm that improves on the previous best approximation by Charikar, Hajiaghayi and Karloff [7]. 2. A sub-exponential time algorithm with much tighter approximation for the case of smooth projection games. 3. A PTAS for planar graphs.National Science Foundation (U.S.) (Grant 1218547

On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in k-uniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lovász [16] gave a k 2 factor LP rounding based approximation, and a matching ( k 2 − o(1)) integrality gap instance was constructed by Aharoni et al. [1]. We prove the following results, which are the first strong hardness results for this problem (here ε> 0 is an arbitrary constant): – NP-hardness of approximating within a factor of ( k 4 − ε) , and – Unique Games-hardness of approximating within a factor of ( k 2 − ε), showing optimality of Lovász’s algorithm under the Unique Games conjecture. The NP-hardness result is based on a reduction from minimum vertex cover in r-uniform hypergraphs for which NP-hardness of approximating within r−1−ε was shown by Dinur et al. [5]. The Unique Games-hardness result is obtained by applying the recent results of Kumar et al. [15], with a slight modification, to the LP integrality gap due to Aharoni et al. [1]. The modification is to ensure that the reduction preserves the desired structural properties of the hypergraph

Inapproximability of hypergraph vertex cover and applications to scheduling problems

Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2¿-¿e for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2¿-¿e for minimizing the makespan in the assembly line problem. These results follow from a new inapproximability result for the Vertex Cover problem on k-uniform hypergraphs that is stronger and simpler than previous results. We show that assuming the UGC, for every k¿=¿2, the problem is inapproximable within k¿-¿e even when the hypergraph is almost k -partite

New and improved bounds for the minimum set cover problem

We study the relationship between the approximation factor for the Set-Cover problem and the parameters Δ : the maximum cardinality of any subset, and k : the maximum number of subsets containing any element of the ground set. We show an LP rounding based approximation of (k−1)(1−e−lnΔk−1)+1 , which is substantially better than the classical algorithms in the range k ≈ ln Δ, and also improves on related previous works [19,22]. For the interesting case when k = θ(logΔ) we also exhibit an integrality gap which essentially matches our approximation algorithm. We also prove a hardness of approximation factor of Ω(logΔ(loglogΔ)2) when k = θ(logΔ). This is the first study of the hardness factor specifically for this range of k and Δ, and improves on the only other such result implicitly proved in [18]