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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.16
URN: urn:nbn:de:0030-drops-163578
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16357/
Beideman, Calvin ;
Chandrasekaran, Karthekeyan ;
Wang, Weihang
Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k
Abstract
We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems - namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph G = (V, E) with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k non-empty parts (V₁, V₂, …, V_k) - known as a k-partition - so as to minimize an objective of interest.
1) If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition.
2) If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut.
We give the first polynomial bound on the number of minmax-k-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an n^{O(k)}p-time deterministic algorithm to enumerate all min-k-cut-sets in hypergraphs, thus improving on the previously known n^{O(k²)}p-time deterministic algorithm, where n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).
BibTeX - Entry
@InProceedings{beideman_et_al:LIPIcs.ICALP.2022.16,
author = {Beideman, Calvin and Chandrasekaran, Karthekeyan and Wang, Weihang},
title = {{Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {16:1--16:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-235-8},
ISSN = {1868-8969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16357},
URN = {urn:nbn:de:0030-drops-163578},
doi = {10.4230/LIPIcs.ICALP.2022.16},
annote = {Keywords: hypergraphs, k-partitioning, counting, enumeration}
}
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Keywords: |
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hypergraphs, k-partitioning, counting, enumeration |
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Collection: |
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49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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28.06.2022 |
