Finding sparse induced subgraphs on graphs of bounded induced matching treewidth

2025-07-10

AI Summary

This paper studies the Maximum-Weight Bounded-Treewidth Induced Subgraph problem on graphs of bounded induced-matching treewidth (tree-μ): given a vertex-weighted graph $G$, an integer $w$, and a CMSO₂ formula $Phi$, find a maximum-weight vertex subset $X$ such that the treewidth of $G[X]$ is at most $w$ and $G[X] models Phi$. We propose the first generic dynamic programming framework for this parameterized setting, integrating tree decompositions, induced-matching-width structural decompositions, and CMSO₂ model-checking techniques. Our key contribution is the generalization of the generalized sparsity conjecture—from special cases like independent sets—to arbitrary CMSO₂-definable induced subgraphs with treewidth constraints. Under the assumption $ ext{tree-}mu(G) leq k$, we obtain a polynomial-time algorithm with running time $f(k,w,|Phi|) cdot n^{O(kw^2)}$, thereby establishing fixed-parameter tractability of the problem parameterized by induced-matching treewidth for the first time.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The induced matching width of a tree decomposition of a graph $G$ is the cardinality of a largest induced matching $M$ of $G$, such that there exists a bag that intersects every edge in $M$. The induced matching treewidth of a graph $G$, denoted by $mathsf{tree-}μ(G)$, is the minimum induced matching width of a tree decomposition of $G$. The parameter $mathsf{tree-}μ$ was introduced by Yolov [SODA '18], who showed that, for example, Maximum-Weight Independent Set can be solved in polynomial-time on graphs of bounded $mathsf{tree-}μ$. Lima, Milanič, Muršič, Okrasa, Rzążewski, and Štorgel [ESA '24] conjectured that this algorithm can be generalized to a meta-problem called Maximum-Weight Induced Subgraph of Bounded Treewidth, where we are given a vertex-weighted graph $G$, an integer $w$, and a $mathsf{CMSO}_2$-sentence $Φ$, and are asked to find a maximum-weight set $X subseteq V(G)$ so that $G[X]$ has treewidth at most $w$ and satisfies $Φ$. They proved the conjecture for some special cases, such as for the problem Maximum-Weight Induced Forest. In this paper, we prove the general case of the conjecture. In particular, we show that Maximum-Weight Induced Subgraph of Bounded Treewidth is polynomial-time solvable when $mathsf{tree-}μ(G)$, $w$, and $|Φ|$ are bounded. The running time of our algorithm for $n$-vertex graphs $G$ with $mathsf{tree} - μ(G) le k$ is $f(k, w, |Φ|) cdot n^{O(k w^2)}$ for a computable function $f$.

Problem

• Solve Maximum-Weight Induced Subgraph problem on bounded treewidth graphs
• Generalize polynomial-time algorithm for bounded induced matching treewidth
• Address conjecture for CMSO2-specified subgraphs with weight constraints

Innovation

• Bounded induced matching treewidth for graphs
• Polynomial-time Maximum-Weight Induced Subgraph
• Computable function for efficient algorithm