Recoverable values for independent sets

Abstract

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 ρ, if for every graph it recovers an independent set of size at least maxis 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) 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 ρ ≥ 1. We design a new randomized algorithm for MIS that ensures an expected recoverable value of at least ρ ≥ 7/3. In addition, we show that approximating MIS in graphs with a given k-coloring within a ratio larger than 2/k is unique games hard. This rules out a natural approach for obtaining ρ ≥ 2.

On percolation and ‐hardness

“…In particular, used nonuniform sampling in order to improve upon the approximation algorithms of for finding the minimum cut in a graph. More recently, sampling subgraphs has been used to find independent sets . The deletion probability of a vertex in the algorithm in is directly proportional to its degree, and there may be dependencies between deletions of different vertices.…”

“…More recently, sampling subgraphs has been used to find independent sets . The deletion probability of a vertex in the algorithm in is directly proportional to its degree, and there may be dependencies between deletions of different vertices.…”

On giant components and treewidth in the layers model

“…The properties of the layers model were used in in designing algorithms for finding large independent sets in graphs. Let α ( G ) denote the size of the maximum independent set in G .…”

“…Observe that α ( T 2 ) can be computed exactly in polynomial time, because T 2 is a forest. In an algorithm was presented for approximating α ( T 3 ) within a ratio of . It is based on the observation that the expected number of edges in T k is at most , and hence the average degree in T 3 is not expected to exceed 2.…”

“…It is based on the observation that the expected number of edges in T k is at most , and hence the average degree in T 3 is not expected to exceed 2. A question that was left open in is whether α ( T 3 ) can be approximated within ratios better than , perhaps even arbitrarily close to 1. This would indeed hold if one could show that T 3 is likely to have small treewidth (sublinear in the number of vertices of T 3 ).…”