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Brubach–Sankararaman–Srinivasan–Xu sampling conjecture for hypergraph matchings

From papers

Let H=(V,E)H=(V,E) be a hypergraph, let w∈R≥0Ew\in\mathbb{R}_{\geq 0}^E be edge weights, and let x∈[0,1]Ex\in[0,1]^E be a fractional matching, meaning that

∑e∈δ(v)x(e)≤1for every v∈V,\sum_{e\in\delta(v)}x(e)\leq 1\qquad\text{for every }v\in V,

where δ(v)=e∈E:v∈e\delta(v)=\\{e\in E:v\in e\\}. Brubach–Sankararaman–Srinivasan–Xu conjecture. It is possible to efficiently sample a matching M⊆EM\subseteq E from a distribution such that every edge e∈Ee\in E satisfies

Pr⁡[e∈M]≥x(e)∣e∣−1+1∣e∣.\Pr[e\in M]\geq\frac{x(e)}{|e|-1+\frac{1}{|e|}}.

This is a distributional and algorithmic strengthening of the Füredi–Kahn–Seymour guarantee. The source describes related weaker bounds and states that the conjecture remains unresolved.

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Sources & referencesView supporting material

Primary source

Georg Anegg, Haris Angelidakis and Rico Zenklusen, “Simpler and Stronger Approaches for Non-Uniform Hypergraph Matching and the Füredi, Kahn, and Seymour Conjecture”, arXiv:2009.00697 (2020).

approximation-algorithmshypergraph-matchingrandomized-algorithms

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