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

Authors Calvin Beideman , Karthekeyan Chandrasekaran , Weihang Wang

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  • Theory of computation → Graph algorithms analysis
Keywords
  • hypergraphs
  • k-partitioning
  • counting
  • enumeration

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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).

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Calvin Beideman, Karthekeyan Chandrasekaran, and Weihang Wang. Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.16

Author Details

Calvin Beideman
  • University of Illinois Urbana-Champaign, IL, USA
Karthekeyan Chandrasekaran
  • University of Illinois Urbana-Champaign, IL, USA
Weihang Wang
  • University of Illinois Urbana-Champaign, IL, USA

Funding

  • Beideman, Calvin: Supported in part by NSF grants CCF-1814613 and CCF-1907937.
  • Chandrasekaran, Karthekeyan: Supported in part by NSF grants CCF-1814613 and CCF-1907937.
  • Wang, Weihang: Supported in part by NSF grants CCF-1814613 and CCF-1907937.

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