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Hypergraph Connectivity Augmentation in Strongly Polynomial Time

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Shubhang Kulkarni

TL;DR

The paper addresses the degree-specified hypergraph connectivity augmentation problem, establishing that feasible hypergraphs can be found with a polynomial number of hyperedges and computed in strongly polynomial time. It develops a general function-cover framework for skew-supermodular and max-max combinations, leveraging LP-based and combinatorial techniques to bound recursion depth and edge counts. The main contributions include strongly polynomial-time algorithms for DS-Hypergraph-CA-using-H, DS-Sim- and near-uniform variants, and several applications to hypergraph-augmentation problems (local, node-to-area, and mixed-hypergraph global connectivity). The results advance exact, scalable methods for hypergraph augmentation, with potential implications for applications in bioinformatics, physics, and machine learning where hypergraph models are natural. The work also raises open questions about improving time to near-linear, and whether solution sizes can be kept linear in all feasible cases, guiding future research directions.

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 show that certain 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. In addition, we develop strongly polynomial time algorithms that return near-uniform hypergraphs as solutions (i.e., every pair of hyperedges differ in size by at most one). As applications of our results, we derive the first strongly polynomial time algorithms for (i) degree-specified hypergraph connectivity augmentation using hyperedges, (ii) degree-specified hypergraph node-to-area connectivity augmentation using hyperedges, and (iii) degree-constrained mixed-hypergraph connectivity augmentation using hyperedges.

Hypergraph Connectivity Augmentation in Strongly Polynomial Time

TL;DR

The paper addresses the degree-specified hypergraph connectivity augmentation problem, establishing that feasible hypergraphs can be found with a polynomial number of hyperedges and computed in strongly polynomial time. It develops a general function-cover framework for skew-supermodular and max-max combinations, leveraging LP-based and combinatorial techniques to bound recursion depth and edge counts. The main contributions include strongly polynomial-time algorithms for DS-Hypergraph-CA-using-H, DS-Sim- and near-uniform variants, and several applications to hypergraph-augmentation problems (local, node-to-area, and mixed-hypergraph global connectivity). The results advance exact, scalable methods for hypergraph augmentation, with potential implications for applications in bioinformatics, physics, and machine learning where hypergraph models are natural. The work also raises open questions about improving time to near-linear, and whether solution sizes can be kept linear in all feasible cases, guiding future research directions.

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 show that certain 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. In addition, we develop strongly polynomial time algorithms that return near-uniform hypergraphs as solutions (i.e., every pair of hyperedges differ in size by at most one). As applications of our results, we derive the first strongly polynomial time algorithms for (i) degree-specified hypergraph connectivity augmentation using hyperedges, (ii) degree-specified hypergraph node-to-area connectivity augmentation using hyperedges, and (iii) degree-constrained mixed-hypergraph connectivity augmentation using hyperedges.
Paper Structure (61 sections, 81 theorems, 85 equations, 1 figure, 2 tables, 3 algorithms)

This paper contains 61 sections, 81 theorems, 85 equations, 1 figure, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Hypergraphs.
  3. Notation.
  4. Degree-Specified Hypergraph Connectivity Augmentation.
  5. Degree-specified skew-supermodular cover problems
  6. Strong Cover Problems.
  7. Weak Cover Problems.
  8. Results
  9. Techniques: Structural and Algorithmic Results
  10. DS-Skew-SupMod-WeakCover-using-H.
  11. DS-Skew-SupMod-WeakCover-using-H with near-uniform hyperedges.
  12. DS-Simul-Skew-SupMod-WeakCover-using-H.
  13. Proof Technique for Theorem \ref{['thm:SzigetiWeakCover:main']}
  14. Algorithm of Szi99.
  15. Our Algorithm.
  16. ...and 46 more sections

Key Result

Theorem 1.1

There exists an algorithm to solve DS-Hypergraph-CA-using-H that runs in time $O(n^7(n + m)^2)$, where $n$ is the number of vertices and $m$ is the number of hypergedges in the input hypergraph. Moreover, if the instance is feasible, then the algorithm returns a solution hypergraph that contains $O(

Figures (1)

  • Figure 1: Subcases in the proof of \ref{['claim:good_vector:swap:constraint(iv)']} for $Z \in \calT_{p", m"}\xspace$.

Theorems & Definitions (174)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.3
  • Theorem 1.4
  • ...and 164 more