Recoverable values for independent sets. Issue 1 (5th March 2013) - Feige, Uriel; Reichman, Daniel
Abstract The notion of recoverable value was advocated in the 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 factor ρ, if for every graph it recovers an independent set of size at least <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcq6q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />maxI∑v∈Imin[1, ρd(v)+1]</alternatives> where d(v) is the degree of vertex v, and I ranges over all independent sets in G. Hence, in a sense, from every vertex v in the maximum independent set the algorithm recovers a value of at least <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcq4m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ρ/(d(v)+1)</alternatives> toward the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. A simple greedy algorithm achieves <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcqfk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ρ≥1</alternatives>. We design a new randomized algorithm for MIS that ensures an expected recoverable factor of at least <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcq9c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ρ≥7/3</alternatives>. In passing, we prove that approximating MIS in graphs with a given k‐coloring within a ratio larger than 2/ k is unique‐games hard. This rules out an alternative approach for obtaining <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcqks" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ρ≥2</alternatives>. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 142–159, 2015