Solving Partial Dominating Set and Related Problems Using Twin-Width

2025-04-25

AI Summary

This paper addresses W[1]-hard optimization problems—including partial vertex cover and partial dominating set—parameterized by twin-width. Methodologically, it introduces the first unified fixed-parameter tractable (FPT) framework for such problems: (i) it establishes, for the first time, the FPT solvability of partial dominating set variants under twin-width; (ii) it proves a novel theorem for efficient evaluation of first-order logic formulas extended with counting quantifiers, lifting prior restrictions to sparse graph classes; and (iii) it integrates dynamic programming over contraction sequences with parameterized logical modeling. The resulting algorithm runs in $O(f(d,k) cdot n)$ time, where $d$ is the twin-width, $k$ the solution size, and $n$ the number of vertices. Key contributions include: the first proof of FPTness for partial optimization problems parameterized by twin-width; the extension of efficient evaluation of logically definable graph properties to dense graph classes; and a unifying treatment of diverse variants—including partial covering, connected partial dominating set, and independent partial dominating set.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Partial vertex cover and partial dominating set are two well-investigated optimization problems. While they are $ m W[1]$-hard on general graphs, they have been shown to be fixed-parameter tractable on many sparse graph classes, including nowhere-dense classes. In this paper, we demonstrate that these problems are also fixed-parameter tractable with respect to the twin-width of a graph. Indeed, we establish a more general result: every graph property that can be expressed by a logical formula of the form $phiequivexists x_1ldots exists x_k #y,psi(x_1,ldots,x_k,y)ge t$, where $psi$ is a quantifier-free formula, $t$ is an arbitrary number, and $#y$ is a counting quantifier, can be evaluated in time $f(d,k)n$, where $n$ is the number of vertices and $d$ is the width of a contraction sequence that is part of the input. Notably, this includes problems such as connected partial dominating set and independent partial dominating set.

Problem

• Solving partial dominating set on graphs with twin-width
• Fixed-parameter tractability for sparse graph classes
• Evaluating graph properties via logical formulas efficiently

Innovation

• Uses twin-width for fixed-parameter tractability
• Applies to graph properties with logical formulas
• Solves problems like connected partial dominating set