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, th...
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Veröffentlicht in: | Mathematical programming 2023, Vol.197 (1), p.27-69 |
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Zusammenfassung: | 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, thus resolving an open problem posed by Guinez and Queyranne (Unpublished manuscript.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. |
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ISSN: | 0025-5610 1436-4646 |
DOI: | 10.1007/s10107-021-01732-0 |