eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
17:1
17:21
10.4230/LIPIcs.APPROX/RANDOM.2020.17
article
Multicriteria Cuts and Size-Constrained k-Cuts in Hypergraphs
Beideman, Calvin
1
Chandrasekaran, Karthekeyan
1
Xu, Chao
2
University of Illinois, Urbana-Champaign, IL, USA
The Voleon Group, Berkeley, CA, USA
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 [Aissi et al., 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 ∈ ℝ^t_+ is O(n²).
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 [Guinez and Queyranne, 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 [Karger, 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.17/LIPIcs.APPROX-RANDOM.2020.17.pdf
Multiobjective Optimization
Hypergraph min-cut
Hypergraph-k-cut