Induced Subgraphs of Bounded Treewidth and the Container Method

SIAM Journal on Computing, Volume 53, Issue 3, Page 624-647, June 2024.
Abstract. A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By [math], we denote a path on [math] vertices. In this paper, we give polynomial-time algorithms for the following problems: the maximum weight independent set problem in long-hole–free graphs and the feedback vertex set problem in [math]-free graphs. Each of the above results resolves a corresponding long-standing open problem. An extended [math] is a five-vertex hole with an additional vertex adjacent to one or two consecutive vertices of the hole. Let [math] be the class of graphs excluding an extended [math] and holes of length at least 6 as induced subgraphs; [math] contains all long-hole–free graphs and all [math]-free graphs. We show that, given an [math]-vertex graph [math] with vertex weights and an integer [math], one can, in time, [math] find a maximum-weight induced subgraph of [math] of treewidth less than [math]. This implies both aforementioned results. To achieve this goal, we extend the framework of potential maximal cliques (PMCs) to containers. Developed by Bouchitté and Todinca [SIAM J. Comput., 31 (2001), pp. 212–232] and extended by Fomin, Todinca, and Villanger [SIAM J. Comput., 44 (2015), pp. 54–87], this framework allows us to solve a wide variety of tasks, including finding a maximum-weight induced subgraph of treewidth less than [math] for fixed [math], in time polynomial in the size of the graph and the number of potential maximal cliques. Further developments, tailored to solve the maximum weight independent set problem within this framework (e.g., for [math]-free [Lokshtanov, Vatshelle, and Villanger, SODA 2014, pp. 570–581] or [math]-free graphs [Grzesik, Klimošová, Pilipczuk, and Pilipczuk, ACM Trans. Algorithms, 18 (2022), pp. 4:1–4:57]), enumerate only a specifically chosen subset of all PMCs of a graph. In all aforementioned works, the final step is an involved dynamic programming algorithm whose state space is based on the considered list of PMCs. Here, we modify the dynamic programming algorithm and show that it is sufficient to consider only a container for each PMC: a superset of the maximal clique that intersects the sought solution only in the vertices of the PMC. This strengthening of the framework not only allows us to obtain our main result but also leads to significant simplifications of the reasoning in previous papers.