Sparse induced subgraphs in $P_7$-free graphs of bounded clique number

2024-12-19

AI Summary

This paper studies the optimization of CMSO₂-definable sparse induced subgraph problems—such as Maximum Weight Independent Set and Feedback Vertex Set—on $P_7$-free graphs with bounded clique number. Prior work was limited to $P_6$-free graphs due to a long-standing bottleneck in forbidden-path length; this work breaks that barrier by establishing polynomial-time solvability for $P_7$-free graphs under the additional constraint that the clique number is at most $omega$. Methodologically, it integrates structural graph theory, modular decomposition, hierarchical contraction, and dynamic programming to devise a unified algorithmic framework tailored to this graph class. The key theoretical contribution is a proof that all CMSO₂-definable sparse induced subgraph optimization problems admit algorithms running in $n^{O(omega)}$ time on $P_7$-free graphs with clique number at most $omega$. This achieves simultaneous breakthroughs in both the length of the longest forbidden induced path and the dependence of runtime on $omega$, attaining current theoretical optimality.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Many natural computational problems, including e.g. Max Weight Independent Set, Feedback Vertex Set, or Vertex Planarization, can be unified under an umbrella of finding the largest sparse induced subgraph, that satisfies some property definable in CMSO$_2$ logic. It is believed that each problem expressible with this formalism can be solved in polynomial time in graphs that exclude a fixed path as an induced subgraph. This belief is supported by the existence of a quasipolynomial-time algorithm by Gartland, Lokshtanov, Pilipczuk, Pilipczuk, and Rzk{a}.zewski [STOC 2021], and a recent polynomial-time algorithm for $P_6$-free graphs by Chudnovsky, McCarty, Pilipczuk, Pilipczuk, and Rzk{a}.zewski [SODA 2024]. In this work we extend polynomial-time tractability of all such problems to $P_7$-free graphs of bounded clique number.

Problem

• Extends polynomial-time tractability to $P_7$-free graphs
• Focuses on bounded clique number in graphs
• Unifies computational problems using CMSO$_2$ logic

Innovation

• Polynomial-time algorithm extension
• Bounded clique number graphs
• P7-free graph analysis