The Price of Being Partial: Complexity of Partial Generalized Dominating Set on Bounded-Treewidth Graphs

2025-06-02

AI Summary

This paper investigates the Partial Generalized Dominating Set problem on graphs of bounded treewidth: given a graph $G$ and integers $k,ell$, determine whether there exists a vertex set $S$ of size at most $k$ such that at most $ell$ vertices violate the $(sigma, ho)$-domination condition. Using tree decompositions and dynamic programming, we design a tight algorithm running in $(s_sigma^p + s_ ho^p + 2)^{mathrm{tw}} cdot |G|^{O(1)}$ time, where $s_sigma^p$ and $s_ ho^p$ are structural parameters intrinsic to the domination constraints $sigma$ and $ ho$. We establish matching algorithmic and complexity lower bounds: via a fine-grained reduction from SETH under Primal Pathwidth, we prove that the base $s_sigma^p + s_ ho^p + 2$ in the exponent is asymptotically optimal. This yields the first complete characterization of the precise time complexity boundary for this problem class, revealing that its computational hardness is fundamentally governed by $s_sigma^p$ and $s_ ho^p$.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• 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
• Security and Privacy: Large-scale security measurements

Abstract

The $(sigma, ho)$-domination framework introduced by Telle [Nord. J. Comput.'94] captures many classical graph problems. For fixed sets $sigma, ho$ of non-negative integers, a $(sigma, ho)$-set of a graph $G$ is a set $S$ such that for every $vin V(G)$, we have (1) if $v in S$, then $|N(v) cap S| in sigma$, and (2) if $v otin S$, then $|N(v) cap S| in ho$. We initiate the study of a natural partial variant of the problem, in which the constraints given by $sigma, ho$ need not be fulfilled for all vertices, but we want to maximize the number of vertices that are happy in the sense that they satisfy (1) or (2) above. Given a graph $G$ and integers $k$ and $ell$, the task of $(sigma, ho)$-MinParDomSet is to decide whether there is a set $S subseteq V(G)$ of size at most $k$ such that at most $ell$ vertices of the graph are not happy under $S$. We consider the problem on graphs of bounded treewidth for nonempty finite or simple cofinite sets $sigma$ and $ ho$, and give matching upper and lower bounds for every such fixed $sigma$ and $ ho$ (under the Primal Pathwidth Strong Exponential Time Hypothesis). Let $s_sigma^ extsf{p} = max sigma + 1$ when $sigma$ is finite, and $min sigma$ when $sigma$ is simple cofinite; define $s_ ho^{ extsf{p}}$ similarly for $ ho$. We show that the problem $(sigma, ho)$-MinParDomSet (1) can be solved in time $(s_sigma^ extsf{p} + s_ ho^{ extsf{p}} + 2)^{tw} cdot |G|^{O(1)}$, when a tree decomposition of width $tw$ is provided together with the input, and (2) for any $varepsilon>0$, no algorithm can exist that solves the problem in time $(s_sigma^ extsf{p} + s_ ho^{ extsf{p}} + 2 - varepsilon)^{pw} cdot |G|^{O(1)}$, even when a path decomposition of width $pw$ is provided together with the input.

Problem

• Studies partial variant of (σ,ρ)-domination on bounded-treewidth graphs.
• Maximizes happy vertices satisfying σ,ρ constraints with size ≤k.
• Provides tight complexity bounds for finite/cofinite σ,ρ under PPSETH.

Innovation

• Partial variant of (σ,ρ)-domination problem
• Bounded treewidth graphs analysis
• Matching upper and lower bounds