Spanning Trees With Edge Conflicts and Wireless Connectivity

TL;DR

This paper presents a new approach to constructing spanning trees in wireless networks that accounts for irregularities and interference, providing approximation algorithms based on the inductive independence of the interference graph.

Contribution

It introduces a novel problem formulation for spanning trees with edge conflicts in wireless environments and offers approximation algorithms with performance guarantees related to the interference graph's properties.

Findings

The problem can be approximated linearly in terms of the inductive independence parameter.

A simple algorithm achieves an $O( ho \, \log n)$-approximation.

Extends results to Steiner trees for multicasting scenarios.

Abstract

We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close --- thus, the available pairs are modeled with a link graph $\mathcal{L}=(V,E)$. Also, signal attenuation need not follow a nice geometric formulas --- hence, interference is modeled by a conflict (hyper)graph $\mathcal{C}=(E,F)$ on the links. The objective is to maximize the efficiency of the communication, or equivalently minimizing the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the interference graph. Specifically, we give a simple algorithm that attains a $O(\rho \log n)$-approximation, where $n$ is the number of nodes and $\rho$ is the inductive independence, and show that near-linear dependence on $\rho$ is also necessary. We also treat an extension to Steiner trees, modeling multicasting, and obtain a comparable result. Our results suggest that several canonical assumptions of geometry, regularity and "niceness" in wireless settings can sometimes be relaxed without a significant hit in algorithm performance.