Tight bounds and conjectures for the isolation lemma

TL;DR

This paper improves lower bounds on the number of isolating weight functions in hypergraphs, proposes a conjecture about extremal cases, and verifies it in specific scenarios, advancing understanding of the Isolation Lemma.

Contribution

It provides new lower bounds for isolating weight functions and introduces a conjecture about the extremal case, with proofs in special cases and asymptotic verification.

Findings

Improved lower bounds match the extremal case asymptotically when M >> n.

Conjecture holds for linear hypergraphs, 1-degenerate hypergraphs, and M=2.

Asymptotic validity of the conjecture when M >> n >> 1.

Abstract

Given a hypergraph $H$ and a weight function $w: V \rightarrow \{1, \dots, M\}$ on its vertices, we say that $w$ is isolating if there is exactly one edge of minimum weight $w(e) = \sum_{i \in e} w(i)$. The Isolation Lemma is a combinatorial principle introduced in Mulmuley et. al (1987) which gives a lower bound on the number of isolating weight functions. Mulmuley used this as the basis of a parallel algorithm for finding perfect graph matchings. It has a number of other applications to parallel algorithms and to reductions of general search problems to unique search problems (in which there are one or zero solutions). The original bound given by Mulmuley et al. was recently improved by Ta-Shma (2015). In this paper, we show improved lower bounds on the number of isolating weight functions, and we conjecture that the extremal case is when $H$ consists of $n$ singleton edges. When $M \gg n$ our improved bound matches this extremal case asymptotically. We are able to show that this conjecture holds in a number of special cases: when $H$ is a linear hypergraph or is 1-degenerate, or when $M = 2$. We also show that it holds asymptotically when $M \gg n \gg 1$.