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

TL;DR

This paper investigates the computational complexity of a partial domination problem on graphs with bounded treewidth, revealing conditions under which the partial variant is harder or equally hard compared to the classical problem.

Contribution

It establishes tight bounds and complexity classifications for the partial $(\sigma, ho)$-domination problem on bounded-treewidth graphs, based on the sets $\sigma$ and $ ho$.

Findings

Matching upper and lower bounds under the Strong Exponential Time Hypothesis.

Partial problem can be as hard as the nonpartial variant for some sets.

Partial problem is significantly harder for certain set choices, like Perfect Code.

Abstract

For fixed sets $\sigma, \rho$ of non-negative integers, the $(\sigma, \rho)$-domination framework introduced by Telle [Nord. J. Comput. 1994] captures many classical graph problems. For a graph $G$, a $(\sigma,\rho)$-set is a set $S$ of vertices such that for every $v\in V(G)$, we have (1) if $v \in S$, then $|N(v) \cap S| \in \sigma$, and (2) if $v \notin S$, then $|N(v) \cap S| \in \rho$. We initiate the study of a natural partial variant $(\sigma,\rho)$-MinParDomSet of the problem, in which the constraints given by $\sigma, \rho$ need not be fulfilled for all vertices, but we want to find a set of size at most $k$ that maximizes the number of vertices that are satisfied in the sense that they satisfy (1) or (2) above. Our goal is to understand whether $(\sigma,\rho)$-MinParDomSet can be solved in the same running time as the nonpartial version, or whether it is strictly harder. Formally, we consider nonempty finite or simple cofinite sets $\sigma$ and $\rho$ (simple cofinite sets are of the form $\mathbb{Z}_{\geq c}$), and we try to determine the smallest constant $c_{\sigma,\rho}$ such that there is a $c_{\sigma,\rho}^{tw}\cdot n^{O(1)}$ time algorithm for the problem if a tree decomposition of width $tw$ is given. We obtain matching upper and lower bounds on $c_{\sigma,\rho}$ for every such fixed $\sigma$ and $\rho$ under the Primal Pathwidth Strong Exponential Time Hypothesis, and establish whether the partial problem is harder than the nonpartial variant. For some sets $\sigma$ and $\rho$, the more general $(\sigma,\rho)$-MinParDomSet has the same complexity as the nonpartial special case (e.g., for Dominating Set), while for other choices, the partial version is significantly harder (e.g., for Perfect Code).