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

Abstract

AbstractA 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.KeywordsBoolean FunctionConnected GraphReduction RulePrimal GraphOrdinary GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.