Maximum weighted independent sets with a budget

TL;DR

This paper studies the Maximum Weighted Budgeted Independent Set problem, focusing on bipartite graphs, proving its NP-hardness and providing a 50% approximation algorithm.

Contribution

It establishes NP-hardness of MWBIS in bipartite graphs and introduces a 50% approximation algorithm matching the LP relaxation gap.

Findings

MWBIS is NP-hard in bipartite graphs.

A 50% approximation algorithm for MWBIS in bipartite graphs.

The approximation matches the LP relaxation integrality gap.

Abstract

Given a graph $G$, a non-negative integer $k$, and a weight function that maps each vertex in $G$ to a positive real number, the \emph{Maximum Weighted Budgeted Independent Set (MWBIS) problem} is about finding a maximum weighted independent set in $G$ of cardinality at most $k$. A special case of MWBIS, when the weight assigned to each vertex is equal to its degree in $G$, is called the \emph{Maximum Independent Vertex Coverage (MIVC)} problem. In other words, the MIVC problem is about finding an independent set of cardinality at most $k$ with maximum coverage. Since it is a generalization of the well-known Maximum Weighted Independent Set (MWIS) problem, MWBIS too does not have any constant factor polynomial time approximation algorithm assuming $P \neq NP$. In this paper, we study MWBIS in the context of bipartite graphs. We show that, unlike MWIS, the MIVC (and thereby the MWBIS) problem in bipartite graphs is NP-hard. Then, we show that the MWBIS problem admits a $\frac{1}{2}$-factor approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation.