Multicriteria cuts and size-constrained k-cuts in hypergraphs

We address counting and optimization variants of multicriteria global min-cut and size-constrained min- k -cut in hypergraphs. For an r -rank n -vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O ( r 2 tr n 3 t - 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 et al. (Math Program 154(1–2):3–28, 2015). In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 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 ( r 2 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 ∈ R ≥ 0 t is O ( n 2 ) . 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sizes is solvable in polynomial-time for constant k , thus resolving an open problem posed by Guinez and Queyranne (Unpublished manuscript. . See also , 2012). Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger (Proceedings of the 4th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 21–30, 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.