Multicritera Cuts and Size-Constrained $k$-cuts in Hypergraphs

TL;DR

This paper advances the understanding of multicriteria min-cuts and size-constrained $k$-cuts in hypergraphs by providing new bounds, enumeration algorithms, and polynomial-time solutions for specific cases, resolving several open problems.

Contribution

It introduces bounds on the number of multiobjective min-cuts, randomized enumeration algorithms, and polynomial-time algorithms for size-constrained $k$-cuts in hypergraphs, extending Karger's random contraction approach.

Findings

Number of multiobjective min-cuts is polynomial for constant rank hypergraphs with fixed criteria.

Enumeration algorithms for all Pareto-optimal cuts run in strongly polynomial time.

Polynomial-time solvability of size-constrained $k$-cut with constant $k$ and lower bounds.

Abstract

We address counting and optimization variants of multicriteria global min-cut and size-constrained min-$k$-cut in hypergraphs. 1. For an $r$-rank $n$-vertex hypergraph endowed with $t$ hyperedge-cost functions, we show that the number of multiobjective min-cuts is $O(r2^{tr}n^{3t-1})$. In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne (Math Programming, 2015). In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 2. We also address node-budgeted multiobjective min-cuts: For an $n$-vertex hypergraph endowed with $t$ vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is $O(r2^{r}n^{t+2})$, where $r$ is the rank of the hypergraph, and the number of node-budgeted $b$-multiobjective min-cuts for a fixed budget-vector $b$ is $O(n^2)$. 3. We show that min-$k$-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant $k$, thus resolving an open problem posed by Queyranne. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger (SODA, 1993). Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained $k$-cuts in hypergraphs.