🤖 AI Summary
This paper resolves the complexity classification of the Maximum Weight Independent Set (MWIS) problem on $H$-free graphs, where every connected component of $H$ is a path or a subdivided claw. It establishes, for the first time, that MWIS admits a quasipolynomial-time algorithm—running in time $|V|^{O(log |V|)}$—on $H$-free graphs for *every* such $H$, thereby completing the dichotomy for $F$-free graphs: all previously open cases are now shown to be tractable. The key technical innovation is a strengthened structural lemma enabling a weight-decreasing extended strip decomposition *without* preprocessing vertex deletions. This is combined with advanced tools from structural graph theory, quasipolynomial branching, balanced separators, and detection of induced $S_{t,t,t}$ subgraphs. The result significantly extends the boundary of efficiently solvable MWIS instances and provides crucial evidence supporting the conjecture that all non-NP-hard $H$-free graph classes lie in P.
📝 Abstract
We show that the Maximum Weight Independent Set problem (MWIS) can be solved in quasi-polynomial time on H-free graphs (graphs excluding a fixed graph H as an induced subgraph) for every H whose every connected component is a path or a subdivided claw (i.e., a tree with at most three leaves). This completes the dichotomy of the complexity of MWIS in F-free graphs for any finite set F of graphs into NP-hard cases and cases solvable in quasi-polynomial time, and corroborates the conjecture that the cases not known to be NP-hard are actually polynomial-time solvable. The key graph-theoretic ingredient in our result is as follows. Fix an integer t ≥ 1. Let St,t,t be the graph created from three paths on t edges by identifying one endpoint of each path into a single vertex. We show that, given a graph G, one can in polynomial time find either an induced St,t,t in G, or a balanced separator consisting of O(log|V(G)|) vertex neighborhoods in G, or an extended strip decomposition of G (a decomposition almost as useful for recursion for MWIS as a partition into connected components) with each particle of weight multiplicatively smaller than the weight of G. This is a strengthening of a result of Majewski, Masařík, Novotná, Okrasa, Pilipczuk, Rzążewski, and Sokołowski [Transactions on Computation Theory 2024] which provided such an extended strip decomposition only after the deletion of O(log|V(G)|) vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above.
