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

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).