New lower bound on max cut of hypergraphs with an application to r-set splitting

• Authors: Archontia C. Giannopoulou;Sudeshna Kolay;Saket Saurabh
• Venue: LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
• Year: 2012

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)={e1,e2,…em}, 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 μH + {n-1}/{r2{r-1}}. Here μH = ∑{i=1}m (1 - 2/2|ei|) = ∑{i=1}m (1 - 2/21-|ei|). We use our result to show that a version of r-Set Splitting, namely, Above Averager-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 μH 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 μH+k hyperedges. We give an algorithm for AA-r-SS that runs in time f(k)nO(1), showing that it is FPT, even when r=c1 logn, for every fixed constant c1r-SS was known to be FPT only for constant r. We also complement our algorithmic result by showing that unless NP ⊆ DTIME(nloglogn), AA-⌈logn ⌉-SS is not in XP.