Covering Problems and Core Percolations on Hypergraphs

TL;DR

This paper generalizes core percolation concepts from graphs to hypergraphs, providing analytical solutions, empirical analysis on real-world data, and improved algorithms for related covering problems.

Contribution

It introduces two hypergraph core percolation models, offers analytical solutions for random hypergraphs, and demonstrates improved algorithms for minimum dominating set problems.

Findings

Hypergraph cores are smaller in real-world data than in random models.

Polynomial-time solvability of hyperedge and vertex cover problems in real-world hypergraphs.

Enhanced algorithm for minimum dominating set problem outperforming existing methods.

Abstract

Covering problems are classical computational problems concerning whether a certain combinatorial structure 'covers' another. For example, the minimum vertex covering problem aims to find the smallest set of vertices in a graph so that each edge is incident to at least one vertex in that set. Interestingly, the computational complexity of the minimum vertex covering problem in graphs is closely related to the core percolation problem, where the core is a special subgraph obtained by the greedy leaf removal procedure. Here, by generalizing the greedy leaf removal procedure in graphs to hypergraphs, we introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for random hypergraphs with arbitrary vertex degree and hyperedge cardinality distributions. We also compute these two cores in several real-world hypergraphs, finding that they tend to be much smaller than their randomized counterparts. This result suggests that both the minimum hyperedge cover problem and the minimum vertex cover problem in those real-world hypergraphs can actually be solved in polynomial time. Finally, we map the minimum dominating set problem in graphs to the minimum hyperedge cover problem in hypergraphs. We show that our generalized greedy leaf removel procedure significantly outperforms the state-of-the-art method in solving the minimum dominating set problem.