Maximum-Weighted $\lambda$-Colorable Subgraph: Revisiting and Applications

Abstract

Given a vertex-weighted graph G and a positive integer $\lambda$, a subset F of the vertices is said to be $\lambda$-colorable if F can be partitioned into at most $\lambda$ independent subsets. This problem of seeking a $\lambda$-colorable F with maximum total weight is known as Maximum-Weighted $\lambda$ -Colorable Subgraph ( $\lambda$ -MWCS). This problem is a generalization of the classical problem Maximum-Weighted Independent Set (MWIS) and has broader applications in wireless networks. All existing approximation algorithms for $\lambda$ -MWCS have approximation bound strictly increasing with $\lambda$. It remains open whether the problem can be approximated with the same factor as the problem MWIS. In this paper, we present new approximation algorithms for $\lambda$ -MWCS. For certain range of $\lambda$, the approximation bounds of our algorithms are the same as those for MWIS, and for a larger range of $\lambda$, the approximation bounds of our algorithms are strictly smaller than the best-known ones in the literature. In addition, we give an exact polynomial-time algorithm for $\lambda$ -MWCS in co-comparability graphs. We also present a number of applications of our algorithms in wireless networking.