Approximating set multi-covers
M\'arton Nasz\'odi, Alexandr Polyanskii

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 , where is the transversal number, is its fractional version, and denotes the maximum degree. We prove for the -fold transversal number . 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 converges to . As a geometric application, we obtain an upper bound on the minimal density of an -fold covering of the -dimensional Euclidean space by translates of any convex body.
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