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Hypergraph Connectivity Augmentation in Strongly Polynomial Time

Authors Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király , Shubhang Kulkarni

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Kristóf Bérczi
  • MTA-ELTE Matroid Optimization Research Group and HUN-REN-ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary
Karthekeyan Chandrasekaran
  • University of Illinois, Urbana-Champaign, IL, USA
Tamás Király
  • MTA-ELTE Matroid Optimization Research Group and HUN-REN-ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary
Shubhang Kulkarni
  • University of Illinois, Urbana-Champaign, IL, USA

Funding

Karthekeyan Chandrasekaran and Shubhang Kulkarni were supported in part by NSF grants CCF-1814613, CCF-1907937, and CCF-2402667. Karthekeyan was supported in part by the Distinguished Guest Scientist Fellowship of the Hungarian Academy of Sciences - grant number VK-6/1/2022. Kristóf and Tamás were supported in part by the Lendület Programme of the Hungarian Academy of Sciences - grant number LP2021-1/2021, by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund - grant number ELTE TKP 2021-NKTA-62 funding scheme, and by the Dynasnet European Research Council Synergy project - grant number ERC-2018-SYG 810115.

Acknowledgements

Part of this work was done while Karthekeyan and Shubhang were visiting Eötvös Loránd University.

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Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni. Hypergraph Connectivity Augmentation in Strongly Polynomial Time. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.22

Abstract

We consider hypergraph network design problems where the goal is to construct a hypergraph that satisfies certain connectivity requirements. For graph network design problems where the goal is to construct a graph that satisfies certain connectivity requirements, the number of edges in every feasible solution is at most quadratic in the number of vertices. In contrast, for hypergraph network design problems, we might have feasible solutions in which the number of hyperedges is exponential in the number of vertices. This presents an additional technical challenge in hypergraph network design problems compared to graph network design problems: in order to solve the problem in polynomial time, we first need to show that there exists a feasible solution in which the number of hyperedges is polynomial in the input size. 
The central theme of this work is to overcome this additional technical challenge for certain hypergraph network design problems. We show that these hypergraph network design problems admit solutions in which the number of hyperedges is polynomial in the number of vertices and moreover, can be solved in strongly polynomial time. Our work improves on the previous fastest pseudo-polynomial run-time for these problems. As applications of our results, we derive the first strongly polynomial time algorithms for (i) degree-specified hypergraph connectivity augmentation using hyperedges and (ii) degree-specified hypergraph node-to-area connectivity augmentation using hyperedges.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypergraphs
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Network optimization
Keywords
  • Hypergraphs
  • Hypergraph Connectivity
  • Submodular Functions
  • Combinatorial Optimization

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