New Lower Bound on Max Cut of Hypergraphs with an Application to r -Set Splitting

Abstract

A classical result by Edwards states that every connected graph G on n vertices and m edges has a cut of size at least $\frac{m}{2}+\frac{n-1}{4}$. We generalize this result to r-hypergraphs, with a suitable notion of connectivity that coincides with the notion of connectivity on graphs for r = 2. More precisely, we show that for every “partition connected” r-hypergraph (every hyperedge is of size at most r) H over a vertex set V(H), and edge set E(H) = {e ₁,e ₂,…e _m }, there always exists a 2-coloring of V(H) with {1, − 1} such that the number of hyperedges that have a vertex assigned 1 as well as a vertex assigned − 1 (or get “split”) is at least $\mu_H+\frac{n-1}{r2^{r-1}}$. Here $\mu_H=\sum_{i=1}^{m}(1- 2/2^{|e_i|})=\sum_{i=1}^{m}(1- 2^{1-|e_i|})$. We use our result to show that a version of r -Set Splitting, namely, Above Average r -Set Splitting (AA- r -SS), is fixed parameter tractable (FPT). Observe that a random 2-coloring that sets each vertex of the hypergraph H to 1 or − 1 with equal probability always splits at least μ _H hyperedges. In AA- r -SS, we are given an r-hypergraph H and a positive integer k and the question is whether there exists a 2-coloring of V(H) that splits at least μ _H  + k hyperedges. We give an algorithm for AA- r -SS that runs in time f(k)n ^O(1), showing that it is FPT, even when r = c ₁ logn, for every fixed constant c ₁ < 1. Prior to our work AA- r -SS was known to be FPT only for constant r. We also complement our algorithmic result by showing that unless NP ⊆ DTIME(n ^loglogn), AA-⌈logn ⌉-SS is not in XP.