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 .
LP+greedy. Consider the following algorithm. Find an optimal half integral solution to the LP, discard all the vertices assigned 0, keep all the vertices assigned 1, and run the greedy algorithm on the graph induced by all vertices that are assigned @math . This algorithm was analyzed for connected graphs. Hochbaum @cite_2 proved an approximation ratio of @math , and Halldorsson and Radhakrishnan @cite_4 (based on their improved analysis of the greedy algorithm) proved an approximation ratio of @math .
@cite_4 @cite_2
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Theminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2) 3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turan's bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved @math performance ratio on graphs with average degree @math , improving on the previous best @math of Hochbaum. Finally, we present an efficient parallel and distributed algorithm attaining the performance guarantees of Greedy. 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.