Tight bounds for budgeted maximum weight independent set in bipartite and perfect graphs

We consider the classic budgeted maximum weight independent set (BMWIS) problem. The input is a graph $G=(V,E)$, where each vertex $v\in V$ has a weight $w\left(v\right)$ and a cost $c\left(v\right)$, and a budget $B$. The goal is to find an independent set $S\subseteq V$ in $G$ such that ${\sum }_{v\in S}c\left(v\right)\le B$, which maximizes the total weight ${\sum }_{v\in S}w\left(v\right)$. Since the problem on general graphs cannot be approximated within ratio ${\left|V\right|}^{1-\varepsilon }$ 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, prior to this work, the best possible approximation guarantees for these graphs were wide open.

In this paper, we give a tight 2-approximation for BMWIS on perfect graphs and bipartite graphs. In particular, we give 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 $w\left(v\right)=c\left(v\right)\forall v\in V$. 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.