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Randomized algorithms to generate hypergraphs with given degree sequences

Michela Ascolese, Matthias Lienau, Matthias Schulte, Anusch Taraz

TL;DR

This paper approaches the question whether there exists a hypergraph whose degrees are equal to a given sequence of integers by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.

Abstract

The question whether there exists a hypergraph whose degrees are equal to a given sequence of integers is a well-known reconstruction problem in graph theory, which is motivated by discrete tomography. In this paper we approach the problem by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.

Randomized algorithms to generate hypergraphs with given degree sequences

TL;DR

This paper approaches the question whether there exists a hypergraph whose degrees are equal to a given sequence of integers by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.

Abstract

The question whether there exists a hypergraph whose degrees are equal to a given sequence of integers is a well-known reconstruction problem in graph theory, which is motivated by discrete tomography. In this paper we approach the problem by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.
Paper Structure (9 sections, 7 theorems, 34 equations, 4 algorithms)

This paper contains 9 sections, 7 theorems, 34 equations, 4 algorithms.

Table of Contents

  1. Introduction and Results
  2. Notation.
  3. History.
  4. Randomized approach.
  5. Our approach.
  6. Results.
  7. Related work.
  8. Algorithms
  9. Proofs

Key Result

Theorem 1

For $n\in\mathbb{N}$ and $k\geq 3$, let $\pi=(d_1,\dots,d_n)$ be a sequence such that $k(k+1)d_{k+2}\leq \sigma$. Then there is a polynomial time randomized algorithm that always returns a $k$-hypergraph $H$ with degree sequence $\pi$ and satisfies

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Definition 5
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 3 more