Min-Max Partitioning of Hypergraphs and Symmetric Submodular Functions

We consider the complexity of minmax partitioning of graphs, hypergraphs and (symmetric) submodular functions. Our main result is an algorithm for the problem of partitioning the ground set of a given symmetric sub modular functionf:2(V)?Rintoknon-empty partsV(1),V-2,...,V-k to minimize max(i)(k)=1f(V-i). Our algorithm runsinn(O)(k(2))Ttime, wheren=|V|andTis the time to evaluatefon a given set; hence,this yields a polynomial time algorithm for any fixedkin the evaluation oracle model.As an immediate corollary, for any fixedk, there is a polynomial-time algorithm forthe problem of partitioning a given hypergraphH=(V,E)intoknon-empty partsto minimize the maximum capacity of the parts. The complexity of this problem,termedMinmax- Hypergraph-k-Part, was raised by Lawler in 1973 (Networks3:275-285, 1973). In contrast to our positive result, the reduction in Chekuri and Li(Theory Comput 16(14):1-8, 2020) implies that whenkis part of the input,Minmax-Hypergraph-k-Partis hard to approximate to within an almost polynomial factorunder the Exponential Time Hypothesis (ETH)