Better Bounds for Incremental Frequency Allocation in Bipartite Graphs

TL;DR

This paper establishes nearly tight bounds on the competitive ratio for incremental frequency allocation in bipartite graphs, improving previous bounds and using combinatorial set family constructions.

Contribution

It provides the first tight bounds for bipartite graphs in incremental frequency allocation, narrowing the ratio to between 1.428 and 1.433.

Findings

Competitive ratio for bipartite graphs is between 1.428 and 1.433.

Improves previous bounds of 4/3 and 1.5.

Uses combinatorial set family constructions for proofs.

Abstract

We study frequency allocation in wireless networks. A wireless network is modeled by an undirected graph, with vertices corresponding to cells. In each vertex we have a certain number of requests, and each of those requests must be assigned a different frequency. Edges represent conflicts between cells, meaning that frequencies in adjacent vertices must be different as well. The objective is to minimize the total number of used frequencies. The offline version of the problem is known to be NP-hard. In the incremental version, requests for frequencies arrive over time and the algorithm is required to assign a frequency to a request as soon as it arrives. Competitive incremental algorithms have been studied for several classes of graphs. For paths, the optimal (asymptotic) ratio is known to be 4/3, while for hexagonal-cell graphs it is between 1.5 and 1.9126. For k-colorable graphs, the ratio of (k+1)/2 can be achieved. In this paper, we prove nearly tight bounds on the asymptotic competitive ratio for bipartite graphs, showing that it is between 1.428 and 1.433. This improves the previous lower bound of 4/3 and upper bound of 1.5. Our proofs are based on reducing the incremental problem to a purely combinatorial (equivalent) problem of constructing set families with certain intersection properties.