The Effect of Sparsity on k-Dominating Set and Related First-Order Graph Properties

We revisit the classic k-Dominating Set problem. Besides its importance as perhaps the most natural W[2]-complete problem, it is among the first problems for which a tight n(k-o(1)) conditional lower bound (for all sufficiently large k), based on the Strong Exponential Time Hypothesis (SETH), was shown (Patrascu and Williams, SODA 2007). Notably, however, the underlying reduction creates dense graphs, raising the question: how much does the sparsity of the graph affect its fine-grained complexity? As our first result, we settle the fine-grained complexity of k-Dominating Set in terms of both the number of nodes n and number of edges m, up to resolving the matrix multiplication exponent omega. Specifically, on the hardness side, we show an mn(k-2-o(1)) lower bound based on SETH, for any dependence of m on n. On the algorithmic side, this is complemented by an mn(k-2+o(1))-time algorithm for all sufficiently large k. For the smallest non-trivial case of k = 2, i.e., 2-Dominating Set, 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)) = O(m(1.187)). If omega = 2, this yields a conditionally tight bound for all k >= 2. To study whether k-Dominating Set is special in its sensitivity to sparsity, we study the effect of sparsity on 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 canonical 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 the monochromatic basic problems, the k-Dominating Set property is the only property 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 +/- 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 >= 2 already on sparse graphs, which is tight for sufficiently large k.