Some remarks on hypergraph matching and the F\"{u}redi-Kahn-Seymour conjecture

TL;DR

This paper proves the F"{u}redi-Kahn-Seymour hypergraph matching conjecture for rank-3 hypergraphs using an iterated rounding algorithm and provides improved bounds for the general case.

Contribution

It confirms the conjecture for rank-3 hypergraphs and introduces a natural iterated rounding algorithm that achieves improved bounds for hypergraph matchings.

Findings

Conjecture is true for rank-3 hypergraphs.

Iterated rounding algorithm achieves improved bounds.

Provides new bounds for the general conjecture.

Abstract

A classic conjecture of F\"{u}redi, Kahn and Seymour (1993) states that given any hypergraph with non-negative edge weights $w(e)$, there exists a matching $M$ such that $\sum_{e \in M} (|e|-1+1/|e|)\, w(e) \geq w^*$, where $w^*$ is the value of an optimum fractional matching. We show the conjecture is true for rank-3 hypergraphs, and is achieved by a natural iterated rounding algorithm. While the general conjecture remains open, we give several new improved bounds. In particular, we show that the iterated rounding algorithm gives $\sum_{e \in M} (|e|-\delta(e))\, w(e) \geq w^*$, where $\delta(e) = |e|/(|e|^2+|e|-1)$, improving upon the baseline guarantee of $\sum_{e \in M} |e|\,w(e) \geq w^*$.