Recoverable values for independent sets

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 leastmax I∑ v ∈ Imin[ 1 , ρ d ( v ) + 1]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 ρ / ( d ( v ) + 1 ) 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 ρ ≥ 1 . We design a new randomized algorithm for MIS that ensures an expected recoverable factor of at least ρ ≥ 7 / 3 . 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 ρ ≥ 2 . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 142–159, 2015