Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time
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2023
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| author | Gartland, Peter Lokshtanov, Daniel Masařík, Tomáš Pilipczuk, Marcin Pilipczuk, Michał Rzążewski, Paweł |
| author_facet | Gartland, Peter Lokshtanov, Daniel Masařík, Tomáš Pilipczuk, Marcin Pilipczuk, Michał Rzążewski, Paweł |
| contents | 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 $\mathcal{F}$-free graphs for any finite set $\mathcal{F}$ 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 $t \geq 1$. Let $S_{t,t,t}$ 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 $S_{t,t,t}$ in $G$, or a balanced separator consisting of $\mathcal{O}(\log |V(G)|)$ 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 [ICALP 2022] which provided such an extended strip decomposition only after the deletion of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_15738 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time Gartland, Peter Lokshtanov, Daniel Masařík, Tomáš Pilipczuk, Marcin Pilipczuk, Michał Rzążewski, Paweł Data Structures and Algorithms 05C69, 05C85 F.2.2; G.2.2 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 $\mathcal{F}$-free graphs for any finite set $\mathcal{F}$ 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 $t \geq 1$. Let $S_{t,t,t}$ 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 $S_{t,t,t}$ in $G$, or a balanced separator consisting of $\mathcal{O}(\log |V(G)|)$ 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 [ICALP 2022] which provided such an extended strip decomposition only after the deletion of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above. |
| title | Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time |
| topic | Data Structures and Algorithms 05C69, 05C85 F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2305.15738 |