Tight Bounds for Budgeted Maximum Weight Independent Set in Bipartite and Perfect Graphs

TL;DR

This paper establishes tight approximation bounds for the budgeted maximum weight independent set problem on bipartite and perfect graphs, providing both upper and lower bounds and exploring related special cases.

Contribution

It presents a tight 2-approximation algorithm for BMWIS on perfect and bipartite graphs and proves a near-matching lower bound based on SSEH, advancing understanding of approximability.

Findings

A 2-approximation for BMWIS on perfect graphs.

A (2 - ε) lower bound for BMWIS on bipartite graphs under SSEH.

Tight bounds for the capacitated maximum weight independent set problem.

Abstract

We consider the classic budgeted maximum weight independent set (BMWIS) problem. The input is a graph $G = (V,E)$, a weight function $w:V \rightarrow \mathbb{R}_{\geq 0}$, a cost function $c:V \rightarrow \mathbb{R}_{\geq 0}$, and a budget $B \in \mathbb{R}_{\geq 0}$. The goal is to find an independent set $S \subseteq V$ in $G$ such that $\sum_{v \in S} c(v) \leq B$, which maximizes the total weight $\sum_{v \in S} w(v)$. Since the problem on general graphs cannot be approximated within ratio $|V|^{1-\varepsilon}$ for any $\varepsilon>0$, BMWIS has attracted significant attention on graph families for which a maximum weight independent set can be computed in polynomial time. Two notable such graph families are bipartite and perfect graphs. BMWIS is known to be NP-hard on both of these graph families; however, the best possible approximation guarantees for these graphs are wide open. In this paper, we give a tight $2$-approximation for BMWIS on perfect graphs and bipartite graphs. In particular, we give We a $(2-\varepsilon)$ lower bound for BMWIS on bipartite graphs, already for the special case where the budget is replaced by a cardinality constraint, based on the Small Set Expansion Hypothesis (SSEH). For the upper bound, we design a $2$-approximation for BMWIS on perfect graphs using a Lagrangian relaxation based technique. Finally, we obtain a tight lower bound for the capacitated maximum weight independent set (CMWIS) problem, the special case of BMWIS where $w(v) = c(v)~\forall v \in V$. We show that CMWIS on bipartite and perfect graphs is unlikely to admit an efficient polynomial-time approximation scheme (EPTAS). Thus, the existing PTAS for CMWIS is essentially the best we can expect.