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 .
Greedy. For MIS, iteratively picking a minimum degree vertex, adding it to an independent set @math and deleting the vertex and its neighbors from the graph is guaranteed to find an independent set of size at least @math @cite_3 @cite_8 . Halldorsson and Radhakrishnan @cite_4 showed that this greedy algorithm produces an independent set of size at least @math (where @math denotes the fraction of vertices in the maximum independent set). For MWIS, weighted greedy that iteratively picks a vertex @math with minimum @math is guaranteed to find an independent set of size at least @math @cite_5 .
@cite_5 @cite_4 @cite_3 @cite_8
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In this paper, we consider three simple and natural greedy algorithms for the maximum weighted independent set problem. We show that two of them output an independent set of weight at least Σv∈V(G) W(v) [d(v) + 1] and the third algorithm outputs an independent set of weight at least Σv∈V(G) W(v)2 [Σu∈NG+(v) W(u)].These results are generalization of theorem of Caro and Wei. 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. Wei discovered that the independence number of a graph G is at least Σv(1 + d(v))−1. It is proved here that if G is a connected triangle-free graph on n ≥ 3 vertices and if G is neither an odd cycle nor an odd path, then the bound above can be increased by nΔ(Δ + 1), where Δ is the maximum degree. This new bound is sharp for even cycles and for three other graphs. These results relate nicely to some algorithms for finding large independent sets. They also have a natural matrix theory interpretation. A survey of other known lower bounds on the independence number is presented.