Recoverable Values for Independent Sets

TL;DR

This paper introduces a new framework for evaluating approximation algorithms for the maximum independent set problem based on recoverable value, and presents a randomized algorithm achieving a recoverable value of at least 7/3.

Contribution

It applies the recoverable value framework to MIS, designs a new randomized algorithm with improved recoverable value, and establishes hardness results related to approximation ratios.

Findings

A randomized algorithm achieves a recoverable value of at least 7/3.

Simple algorithms guarantee a recoverable value of at least 1.

Approximating MIS within certain ratios is shown to be hard under the unique games conjecture.

Abstract

The notion of {\em 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 {\em recoverable value} $\rho$, if for every graph it recovers an independent set of size at least $\max_I \sum_{v\in I} \min[1,\rho/(d(v) + 1)]$, 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 $\rho/(d_v + 1)$ 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 $\rho \ge 1$. We design a new randomized algorithm for MIS that ensures an expected recoverable value of at least $\rho \ge 7/3$. 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 $\rho \ge 2$.