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.
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.
Multiobjective Optimization
Hypergraph min-cut
Hypergraph-k-cut
Mathematics of computing~Combinatorial optimization
17:1-17:21
RANDOM
http://arxiv.org/abs/2006.11589
Calvin
Beideman
Calvin Beideman
University of Illinois, Urbana-Champaign, IL, USA
Supported in part by NSF CCF-1907937.
Karthekeyan
Chandrasekaran
Karthekeyan Chandrasekaran
University of Illinois, Urbana-Champaign, IL, USA
Supported in part by NSF CCF-1907937 and CCF-1814613.
Chao
Xu
Chao Xu
The Voleon Group, Berkeley, CA, USA
10.4230/LIPIcs.APPROX/RANDOM.2020.17
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Calvin Beideman, Karthekeyan Chandrasekaran, and Chao Xu
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