New lower bound on Max Cut of hypergraphs with an application to r -set splitting

A classical result by Edwards states that every connected graph G on n vertices and m edges has a cut of size at least m/2 + 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 μ _H + n-1/r2 ^r-1. Here μ _H = ^m _i=1 (1-2/2 ^|ei|) = ^m _i=1 (1-2 ^1-|ei|). 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 κ 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(κ)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 ^{log log n} ), AA-⌈logn⌉-SS is not in XP.