Approximating set multi-covers

TL;DR

This paper extends classical bounds on hypergraph transversals to $f$-fold covers, providing new theoretical limits and a greedy algorithm approach, with applications in combinatorics and geometry.

Contribution

It introduces a bound on the $f$-fold transversal number related to fractional transversals, achieved via a greedy algorithm, and applies this to geometric covering problems.

Findings

Bound on $ au_f$ in terms of $ au^*$, $ ext{ln}\Delta$, and $f$

Greedy algorithm achieves the bound non-probabilistically

Estimate on convergence rate of $ au_f/f$ to $ au^*$

Abstract

Johnson and Lov\'asz and Stein proved independently that any hypergraph satisfies $\tau\leq (1+\ln \Delta)\tau^{\ast}$, where $\tau$ is the transversal number, $\tau^{\ast}$ is its fractional version, and $\Delta$ denotes the maximum degree. We prove $\tau_f\leq c \tau^{\ast}\max\{\ln \Delta, f\}$ for the $f$-fold transversal number $\tau_f$. Similarly to Johnson, Lov\'asz and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm. As a combinatorial application, we prove an estimate on how fast $\tau_f/f$ converges to $\tau^{\ast}$. As a geometric application, we obtain an upper bound on the minimal density of an $f$-fold covering of the $d$-dimensional Euclidean space by translates of any convex body.