Cover-Decomposition and Polychromatic Numbers

TL;DR

This paper explores the cover-decomposition and polychromatic numbers in hypergraphs, providing new algorithms and bounds beyond geometric cases, and discusses applications to sensor coverage.

Contribution

It introduces algorithms with near-tight bounds for hypergraphs with bounded hyperedge size, paths in trees, and bounded VC-dimension, expanding the scope beyond geometric hypergraphs.

Findings

Algorithms yield near-tight bounds for specific hypergraph families

Discrepancy theory and linear programming are effective tools

Discussion on extending cover-decomposition to sensor coverage

Abstract

A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge-covers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded VC-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover.