Tight Bounds for Budgeted Maximum Weight Independent Set in Bipartite and Perfect Graphs
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We consider the classic budgeted maximum weight independent set (BMWIS) problem. The input is a graph G = (V,E), a weight function w:V →ℝ_≥ 0, a cost function c:V →ℝ_≥ 0, and a budget B ∈ℝ_≥ 0. The goal is to find an independent set S ⊆ V in G such that ∑_v ∈ S c(v) ≤ B, which maximizes the total weight ∑_v ∈ S w(v). Since the problem on general graphs cannot be approximated within ratio |V|^1-ε for any ε>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-ε) 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) ∀ v ∈ 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.
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