Channel allocation revisited through 1-extendability of graphs

TL;DR

The paper tackles channel allocation in Wi-Fi by introducing the 1-extendable chromatic number $\chi_{\text{1-ext}}(G)$, which partitions a conflict graph into parts each inducing a 1-extendable graph to avoid starvation under saturation. It develops modular-decomposition based algorithms and structural results, achieving a single-exponential-time 1-extendability test in modular-width and a $\alpha(G)^{O(mw(G)k)}$-time scheme for constructing a 1-extendable $k$-partition, with a specialized, quasi-polynomial solution for cographs. Extremal analysis yields a general upper bound $\chi_{\text{1-ext}}(G) \le 2\sqrt{n}$ and a tight logarithmic bound $\chi_{\text{1-ext}}(G) \le \log_2(\alpha(G))+1$ for cographs, along with concrete lower-bound constructions using complete multipartite and interval graphs. The results bridge graph-theoretic properties with practical wireless networking considerations, guiding fair and scalable channel allocations while highlighting open questions for polynomial-time solutions in restricted graph classes and geometric graph models.

Abstract

We revisit the classical problem of channel allocation for Wi-Fi access points (AP). Using mechanisms such as the CSMA/CA protocol, Wi-Fi access points which are in conflict within a same channel are still able to communicate to terminals. In graph theoretical terms, it means that it is not mandatory for the channel allocation to correspond to a proper coloring of the conflict graph. However, recent studies suggest that the structure -- rather than the number -- of conflicts plays a crucial role in the performance of each AP. More precisely, the graph induced by each channel must satisfy the so-called $1$-extendability property, which requires each vertex to be contained in an independent set of maximum cardinality. In this paper we introduce the 1-extendable chromatic number, which is the minimum size of a partition of the vertex set of a graph such that each part induces a 1-extendable graph. We study this parameter and the related optimization problem through different perspectives: algorithms and complexity, structure, and extremal properties. We first show how to compute this number using modular decompositions of graphs, and analyze the running time with respect to the modular width of the input graph. We also focus on the special case of cographs, and prove that the 1-extendable chromatic number can be computed in quasi-polynomial time in this class. Concerning extremal results, we show that the 1-extendable chromatic number of a graph with $n$ vertices is at most $2\sqrt{n}$, whereas the classical chromatic number can be as large as $n$. We are also able to construct graphs whose 1-extendable chromatic number is at least logarithmic in the number of vertices.

Figures (3)

• Figure 1: Two graphs and the $p_v$ value of each vertex, obtained with ns-3 simulations. ns-3 is a network simulator that implements the Wi-Fi protocol in the same way as the standards, and incorporates a realistic radio layer. The simulated Wi-Fi corresponds to the IEEE 802.11ax amendment with frame aggregation enabled and a physical transmission rate of 144.4Mbit/s. The top graph is $1$-extendable and thus each vertex has a non-negligible throughput. The bottom graph is not $1$-extendable, and the middle vertex has a very low throughput.
• Figure 2: A complete multipartite graph (left) and a $1$-extendable $3$-partition of it, represented as a coloring (right). For the sake of readability, the edges are not shown, the rectangles represent the partition into independent sets.
• Figure 3: An interval representation of $G_4$, and a 1-extendable $4$-partition of it. In the partition, each color class induces a disjoint union of cliques, which is $1$-extendable.

Theorems & Definitions (36)

• theorem thmcountertheorem
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• proposition thmcounterproposition
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• theorem thmcountertheorem
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• lemma thmcounterlemma
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• lemma thmcounterlemma
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