Conditionally Tight Algorithms for Maximum k-Coverage and Partial k-Dominating Set via Arity-Reducing Hypercuts

TL;DR

This paper develops new algorithms and tight bounds for the Maximum $k$-Coverage and Partial $k$-Dominating Set problems, improving runtime efficiency and establishing optimality under certain complexity hypotheses.

Contribution

It introduces algorithms with near-optimal running times for Partial $k$-Dominating Set and Maximum $k$-Coverage, based on parameters like universe size, set size, and frequency, and proves their conditional optimality.

Findings

Algorithms achieve near-optimal runtime bounds under complexity hypotheses.

Matching upper and lower bounds established for sparse graphs.

Runtime depends on parameters like universe size, set size, and frequency.

Abstract

We revisit the classic Maximum $k$-Coverage problem: Determine the largest number $t$ of elements that can be covered by choosing $k$ sets from a given family $\mathcal{F} = \{S_1,\dots, S_n\}$ of a size-$u$ universe. A notable special case is Partial $k$-Dominating Set, where one chooses $k$ vertices in a graph to maximize the number of dominated vertices. Extensive research has established strong hardness results for various aspects of Maximum $k$-Coverage, such as tight inapproximability results, $W[2]$-hardness, and a conditionally tight worst-case running time of $n^{k\pm o(1)}$. In this paper we ask: (1) Can this time bound be improved for small $t$, at least for Partial $k$-Dominating Set, ideally to time~$t^{k\pm O(1)}$? (2) More ambitiously, can we even determine the best-possible running time of Maximum $k$-Coverage with respect to the perhaps most natural parameters: the universe size $u$, the maximum set size $s$, and the maximum frequency $f$? We successfully resolve both questions. (1) We give an algorithm that solves Partial $k$-Dominating Set in time $O(nt + t^{\frac{2\omega}{3} k+O(1)})$ if $\omega \ge 2.25$ and time $O(nt+ t^{\frac{3}{2} k+O(1)})$ if $\omega \le 2.25$, where $\omega \le 2.372$ is the matrix multiplication exponent. From this we derive a time bound that is conditionally optimal, regardless of $\omega$, based on the well-established $k$-clique and 3-uniform hyperclique hypotheses from fine-grained complexity. We also obtain matching upper and lower bounds for sparse graphs. To address (2) we design an algorithm for Maximum $k$-Coverage running in time $$ \min \left\{ (f\cdot \min\{\sqrt[3]{u}, \sqrt{s}\})^k + \min\{n,f\cdot \min\{\sqrt{u}, s\}\}^{k\omega/3}, n^k\right\} \cdot g(k)n^{\pm O(1)}, $$ and, surprisingly, further show that this complicated time bound is also conditionally optimal.