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.

翻译：我们考虑经典预算最大权独立集（BMWIS）问题。输入为一个图 $G = (V,E)$、权函数 $w:V \rightarrow \mathbb{R}_{\geq 0}$、成本函数 $c:V \rightarrow \mathbb{R}_{\geq 0}$ 以及预算 $B \in \mathbb{R}_{\geq 0}$。目标是在 $G$ 中找到一个独立集 $S \subseteq V$，使得 $\sum_{v \in S} c(v) \leq B$，并最大化总权 $\sum_{v \in S} w(v)$。由于一般图上的该问题对于任意 $\varepsilon>0$ 无法在比例 $|V|^{1-\varepsilon}$ 内近似，BMWIS 在那些可在多项式时间内计算最大权独立集的图族上引起了广泛关注。两个显著的图族是二分图和完美图。已知 BMWIS 在这两个图族上均是 NP-难的；然而，这些图上的最佳可能近似保证仍悬而未解。在本文中，我们给出了完美图和二分图上 BMWIS 的一个紧的 $2$-近似。具体而言，基于小集扩张假设（SSEH），即使在预算被替换为基数约束的特殊情况下，我们给出了二分图上 BMWIS 的一个 $(2-\varepsilon)$ 下界。对于上界，我们利用拉格朗日松弛技术为完美图上的 BMWIS 设计了一个 $2$-近似。最后，我们获得了带容量限制的最大权独立集（CMWIS）问题（即 BMWIS 中 $w(v) = c(v)~\forall v \in V$ 的特例）的紧下界。我们证明，二分图和完美图上的 CMWIS 不太可能允许高效的多项式时间近似方案（EPTAS）。因此，现有的 CMWIS 的 PTAS 本质上是我们所能期待的最佳结果。