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

2026-01-23

AI Summary

This work investigates conditionally optimal algorithms for Maximum k-Coverage and its special case, Partial k-Dominating Set. Leveraging parameters such as the number of covered elements \( t \), universe size \( u \), maximum set size \( s \), and maximum frequency \( f \), the study introduces refined algorithms based on arity-reducing hypercuts. It presents the first conditionally tight algorithm for Partial k-Dominating Set parameterized by \( t \), achieving running times of \( O(nt + t^{(2\omega/3)k + O(1)}) \) or \( O(nt + t^{(3/2)k + O(1)}) \), where \( \omega \) denotes the matrix multiplication exponent. For Maximum k-Coverage, the paper establishes conditionally optimal time upper bounds under various parameter combinations. The analysis relies on the k-clique and 3-uniform hyperclique hypotheses, combined with hypergraph pruning techniques, to deliver a fine-grained characterization of computational complexity.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Maximum k-Coverage
• Partial k-Dominating Set
• fine-grained complexity
• parameterized algorithms
• conditional lower bounds

Innovation

• Maximum k-Coverage
• Partial k-Dominating Set
• fine-grained complexity
• conditionally tight algorithms
• arity-reducing hypercuts