Distributed Online Frequency Assignment in Cellular Networks

Multicoloring of cannonball graphs

“…As this is the first result for the multicoloring problem of cannonball graphs, we believe that further improvements can be done. Among the interesting problems that remain open are: improving of the competitive ratio 11/6, finding some distributed algorithms for multicoloring cannonball graphs, or finding some k-local algorithms for some k, similarly as in 2-dimensional case for hexagonal graphs (for definition of k-local algorithms see [10]). We already mentioned that in the 2-dimensional case, better bounds were obtained for triangle-free hexagonal graphs.…”

“…McDiarmid and Reed proved in [12] that multicoloring of hexagonal graphs is NP-complete. In the last decade there were several results on upper bounds for the multichromatic number in terms of weighted clique number for hexagonal graphs, some of which also provide approximation algorithms that are fully distributed and run in constant time [8,9,10,12,13,14,16,18,19,20,15,21,23,24,25,27]. The best known approximation ratios are χ m (G) ≤ (4/3)ω(G) + O(1) in general [12,14,18] and χ m (G) ≤ (7/6)ω(G)+O(1) for triangle free hexagonal graphs [8,15,16] [12].…”

A Self-Managed Distributed Channel Selection Algorithm for WLANs

“…Since the channel assignment problem is NP-hard, numerous heuristic schemes have been proposed that come with few, if any, guarantees on performance. Notable exceptions include Sparl and Zerovnik [12], Sudeep and Vishwanathan [13], Janssen et al [14] and Narayanan and Shende [15] who study distributed algorithms for frequency assignment in cellular networks and provide competitive performance bounds. However, this work concentrates on the cellular network case where the graph is always a subgraph of the triangular lattice.…”

A 1-Local Asymptotic 13/9-Competitive Algorithm for Multicoloring Hexagonal Graphs

“…In this paper, we focus on distributed algorithms for the multicoloring, i.e., each vertex is an independent server, which runs the algorithm to assign multicolors to the vertex based on what is known as k-local information. The concept of k-local distributed algorithms was introduced by Janssen et al [8], where an algorithm is k-local if the computation at a vertex depends only on the information of the neighboring vertices of at most k distance away (suppose each edge has unit distance). Similar to frequency allocation problem, we can assume that in multicoloring problem, each vertex also knows its position in the graph.…”

“…In [8], Janssen et al proved (the next lemma) that a k-local c-approximate off-line algorithm can be easily converted to a k-local c-competitive online algorithm. Thus, to design a k-local online algorithm, we need only to focus on the k-local off-line problem.…”