Maximum-Weighted $$\lambda $$ λ -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.