Fine-Grained Complexity of Multiple Domination and Dominating Patterns in Sparse Graphs

TL;DR

This paper investigates the fine-grained complexity of various domination problems in sparse graphs, providing conditionally optimal algorithms and complexity bounds based on popular hypotheses.

Contribution

It introduces new algorithms and complexity results for multiple domination variants, improving understanding of their computational hardness in sparse graphs.

Findings

Conditional optimal algorithms for r-Multiple k-Dominating Set.

Complexity bounds for H-Dominating Set variants like Dominating Clique.

Demonstrates larger speed-ups for natural domination variants in sparse graphs.

Abstract

The study of domination in graphs has led to a variety of domination problems studied in the literature. Most of these follow the following general framework: Given a graph $G$ and an integer $k$, decide if there is a set $S$ of $k$ vertices such that (1) some inner property $\phi(S)$ (e.g., connectedness) is satisfied, and (2) each vertex $v$ satisfies some domination property $\rho(S, v)$ (e.g., there is an $s\in S$ that is adjacent to $v$). Since many real-world graphs are sparse, we seek to determine the optimal running time of such problems in both the number $n$ of vertices and the number $m$ of edges in $G$. While the classic dominating set problem admits a rather limited improvement in sparse graphs (Fischer, K\"unnemann, Redzic SODA'24), we show that natural variants studied in the literature admit much larger speed-ups, with a diverse set of possible running times. Specifically, we obtain conditionally optimal algorithms for: 1) $r$-Multiple $k$-Dominating Set (each vertex must be adjacent to at least $r$ vertices in $S$): If $r\le k-2$, we obtain a running time of $(m/n)^{r} n^{k-r+o(1)}$ that is conditionally optimal assuming the 3-uniform hyperclique hypothesis. In sparse graphs, this fully interpolates between $n^{k-1\pm o(1)}$ and $n^{2\pm o(1)}$, depending on $r$. Curiously, when $r=k-1$, we obtain a randomized algorithm beating $(m/n)^{k-1} n^{1+o(1)}$ and we show that this algorithm is close to optimal under the $k$-clique hypothesis. 2) $H$-Dominating Set ($S$ must induce a pattern $H$). We conditionally settle the complexity of three such problems: (a) Dominating Clique ($H$ is a $k$-clique), (b) Maximal Independent Set of size $k$ ($H$ is an independent set on $k$ vertices), (c) Dominating Induced Matching ($H$ is a perfect matching on $k$ vertices).