The notion of recoverable value was advocated in work of Feige, Immorlica, Mirrokni and Nazerzadeh [Approx 2009] as a measure of quality for approximation algorithms. There this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has recoverable value @math , if for every graph it recovers an independent set of size at least @math , where @math is the degree of vertex @math , and @math ranges over all independent sets in @math . Hence, in a sense, from every vertex @math in the maximum independent set the algorithm recovers a value of at least @math towards the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. It easily follows from known results that some simple algorithms for MIS ensure @math . We design a new randomized algorithm for MIS that ensures an expected recoverable value of at least @math . In addition, we show that approximating MIS in graphs with a given @math -coloring within a ratio larger than @math is unique games hard. This rules out a natural approach for obtaining @math .
We study the inapproximability of Vertex Cover and Independent Set on degree @math graphs. We prove that: Vertex Cover is Unique Games-hard to approximate to within a factor @math . This exactly matches the algorithmic result of Halperin halperin02improved up to the @math term. Independent Set is Unique Games-hard to approximate to within a factor @math . This improves the @math Unique Games hardness result of Samorodnitsky and Trevisan samorodnitsky06gowers . Additionally, our result does not rely on the construction of a query efficient PCP as in samorodnitsky06gowers .  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 Lovasz [16] gave a k 2 factor LP rounding based approximation, and a matching (k 2 - o(1)) integrality gap instance was constructed by [1]. We prove the following results, which are the first strong hardness results for this problem (heree > 0 is an arbitrary constant): - NP-hardness of approximating within a factor of (k 4 -e), and - Unique Games-hardness of approximating within a factor of (k 2 -e), showing optimality of Lovasz'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-e was shown by [5]. The Unique Games-hardness result is obtained by applying the recent results of [15], with a slight modification, to the LP integrality gap due to [1]. The modification is to ensure that the reduction preserves the desired structural properties of the hypergraph. Abstract In this paper we describe a collection of efficient algorithms that deliver approximate solution to the weighted stable set, vertex cover and set packing problems. All algorithms guarantee bounds on the ratio of the heuristic solution to the optimal solution.