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The Structure of Hypergraphs Arising in Cellular Mobile Communication Systems

Ashwin Ganesan

TL;DR

It is shown that the problem of computing the interference degree of a hypergraph is NP-hard and some properties and results concerning this hypergraph invariant are proved.

Abstract

An assumption that researchers have often used to model interference in a wireless network is the unit disk graph model. While many theoretical results and performance guarantees have been obtained under this model, an open research direction is to extend these results to hypergraph interference models. Motivated by recent results that the worst-case performance of the distributed maximal scheduling algorithm is characterized by the interference degree of the hypergraph, in the present work we investigate properties of the interference degree of the hypergraph and the structure of hypergraphs arising from physical constraints. We show that the problem of computing the interference degree of a hypergraph is NP-hard and we prove some properties and results concerning this hypergraph invariant. We investigate which hypergraphs are realizable, i.e. which hypergraphs arise in practice, based on physical constraints, as the interference model of a wireless network. In particular, a question that arises naturally is: what is the maximal value of $r$ such that the hypergraph $K_{1,r}$ is realizable? We determine this quantity for various integral and nonintegral values of the path loss exponent of signal propagation. We also investigate hypergraphs generated by line networks.

The Structure of Hypergraphs Arising in Cellular Mobile Communication Systems

TL;DR

It is shown that the problem of computing the interference degree of a hypergraph is NP-hard and some properties and results concerning this hypergraph invariant are proved.

Abstract

An assumption that researchers have often used to model interference in a wireless network is the unit disk graph model. While many theoretical results and performance guarantees have been obtained under this model, an open research direction is to extend these results to hypergraph interference models. Motivated by recent results that the worst-case performance of the distributed maximal scheduling algorithm is characterized by the interference degree of the hypergraph, in the present work we investigate properties of the interference degree of the hypergraph and the structure of hypergraphs arising from physical constraints. We show that the problem of computing the interference degree of a hypergraph is NP-hard and we prove some properties and results concerning this hypergraph invariant. We investigate which hypergraphs are realizable, i.e. which hypergraphs arise in practice, based on physical constraints, as the interference model of a wireless network. In particular, a question that arises naturally is: what is the maximal value of such that the hypergraph is realizable? We determine this quantity for various integral and nonintegral values of the path loss exponent of signal propagation. We also investigate hypergraphs generated by line networks.
Paper Structure (18 sections, 33 theorems, 46 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 18 sections, 33 theorems, 46 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Unit Disk Graphs
  4. Signal Propagation Model and Path Loss Exponents
  5. The Hypergraph of a Wireless Network
  6. A Distributed Maximal Scheduling Algorithm and The Interference Degree of a Hypergraph
  7. Notation and Terminology
  8. Related Work
  9. The Interference Degree of Structured Hypergraphs
  10. Non-Realizable Hypergraphs: Preliminaries
  11. Nonrealizability of the Hypergraph $K_{1,5}$ When $\gamma=4$
  12. Nonrealizability of the Hypergraph $K_{1,5}$ When $\gamma=3$
  13. Nonrealizability of the Hypergraph $K_{1,4}$ When $\gamma \le 2$
  14. Nonrealizability of the Hypergraph $K_{1,3}$ When $\gamma \le 1$
  15. Extensions to Non-integral Values of the Path Loss Exponent $\gamma$
  16. ...and 3 more sections

Key Result

Theorem 2

Li:Negi:2012 Let $H$ be a hypergraph, with $\Delta_{ij} = \Delta_{ij}(H)$ as defined above. A sufficient condition for a demand vector $\tau$ to be feasible is that

Figures (1)

  • Figure 1: Five stations on the unit circle, with stations $s_1,s_2$ and $s_5$ close enough to each other that $\delta_1+\delta_5=2\pi/15$. Relevant to the proof of Theorem \ref{['thm:K1:5:alpha:4']}.

Theorems & Definitions (36)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 26 more