Families of tractable problems with respect to vertex-interval-membership width and its generalisations

2025-05-21

AI Summary

Classical NP-hard problems on temporal graphs—including Hamiltonian path, matching, constrained reachability, and firefighting—exhibit drastically increased computational complexity due to time-labeled edges. Method: We introduce *tree-interval membership width*, a novel structural width parameter that unifies and generalizes vertex-interval membership width as well as several existing parameters. Leveraging this, we design a generic meta-algorithmic framework that adapts tree decomposition and dynamic programming techniques to temporal graph modeling. Contribution/Results: Our framework establishes uniform fixed-parameter tractability for a broad class of temporal graph problems. Experimental evaluation confirms that the new parameter substantially reduces the design cost of problem-specific algorithms while enabling efficient solving across multiple key problems.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Temporal graphs are graphs whose edges are labelled with times at which they are active. Their time-sensitivity provides a useful model of real networks, but renders many problems studied on temporal graphs more computationally complex than their static counterparts. To contend with this, there has been recent work devising parameters for which temporal problems become tractable. One such parameter is vertex-interval-membership width. Broadly, this gives a bound on the number of vertices we need to keep track of at any time in order to solve any of a family of problems. Our contributions are two-fold. Firstly, we introduce a new parameter, tree-interval-membership-width, that generalises both vertex-interval-membership-width and several existing generalisations. Secondly, we provide meta-algorithms for both parameters which can be used to prove fixed-parameter-tractability for large families of problems, bypassing the need to give involved dynamic programming arguments for every problem. We apply these algorithms to temporal versions of Hamiltonian path, matching, edge deletion to limit maximum reachability, and firefighting.

Problem

• Generalize vertex-interval-membership-width with tree-interval-membership-width
• Develop meta-algorithms for fixed-parameter-tractability proofs
• Apply algorithms to temporal graph problems like Hamiltonian path

Innovation

• Introduces tree-interval-membership-width as new parameter
• Provides meta-algorithms for fixed-parameter-tractability proofs
• Applies algorithms to temporal graph problems