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### Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k

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### 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},
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