We revisit $k$-Dominating Set, one of the first problems for which a tight $n^k-o(1)$ conditional lower bound (for $k\ge 3$), based on SETH, was shown (P\u{a}tra\c{s}cu and Williams, SODA 2007). However, the underlying reduction creates dense graphs, raising the question: how much does the sparsity of the graph affect its fine-grained complexity? We first settle the fine-grained complexity of $k$-Dominating Set in terms of both the number of nodes $n$ and number of edges $m$. Specifically, we show an $mn^{k-2-o(1)}$ lower bound based on SETH, for any dependence of $m$ on $n$. This is complemented by an $mn^{k-2+o(1)}$-time algorithm for all $k\ge 3$. For the $k=2$ case, we give a randomized algorithm that employs a Bloom-filter inspired hashing to improve the state of the art of $n^{\omega+o(1)}$ to $m^{\omega/2+o(1)}$. If $\omega=2$, this yields a conditionally tight bound for all $k\ge 2$. To study if $k$-Dominating Set is special in its sensitivity to sparsity, we consider a class of very related problems. The $k$-Dominating Set problem belongs to a type of first-order definable graph properties that we call monochromatic basic problems. These problems are the natural monochromatic variants of the basic problems that were proven complete for the class FOP of first-order definable properties (Gao, Impagliazzo, Kolokolova, and Williams, TALG 2019). We show that among these problems, $k$-Dominating Set is the only one whose fine-grained complexity decreases in sparse graphs. Only for the special case of reflexive properties, is there an additional basic problem that can be solved faster than $n^{k\pm o(1)}$ on sparse graphs. For the natural variant of distance-$r$ $k$-dominating set, we obtain a hardness of $n^{k-o(1)}$ under SETH for every $r\ge 2$ already on sparse graphs, which is tight for sufficiently large $k$.

翻译：我们重新审视$k$-支配集问题，这是首个基于SETH证明具有紧条件$n^{k-o(1)}$下界（$k\ge 3$）的问题之一（Pătrașcu和Williams，SODA 2007）。然而，其底层归约构造了稠密图，这引发了一个问题：图的稀疏性在多大程度上影响其细粒度复杂度？我们首先从节点数$n$和边数$m$两个维度确定了$k$-支配集的细粒度复杂度。具体而言，我们基于SETH证明了对$m$的任何依赖关系均存在$mn^{k-2-o(1)}$的下界，并给出对所有$k\ge 3$的$mn^{k-2+o(1)}$时间算法与之匹配。对于$k=2$的情形，我们提出一种采用布隆过滤器启发式哈希的随机算法，将现有最优的$n^{\omega+o(1)}$改进为$m^{\omega/2+o(1)}$。若$\omega=2$，则对所有$k\ge 2$产生条件紧界。为探究$k$-支配集对稀疏性的敏感性是否具有特殊性，我们考虑一类密切相关的问题。$k$-支配集问题属于可定义的一阶图性质，我们称之为单色基本问题。这些问题是基本问题的自然单色变体，已被证明对一阶可定义性质类FOP具有完备性（Gao、Impagliazzo、Kolokolova和Williams，TALG 2019）。我们证明，在这些问题中，$k$-支配集是唯一在稀疏图上细粒度复杂度降低的问题。仅对自反性质的特殊情形，存在另一个基本问题可在稀疏图上以快于$n^{k\pm o(1)}$的时间求解。对于距离$r$的$k$-支配集这一自然变体，我们证明即使在稀疏图上，对每个$r\ge 2$，在SETH假设下其困难度为$n^{k-o(1)}$，该界对充分大的$k$是紧的。