51,876 research outputs found
Channel Assignment with Separation on Trees and Interval Graphs
Given a vector of non increasing positive integers, and an undirected graph , an -coloring of is a function from the vertex set to a set of nonnegative integers such that , if where is the distance (i.e. the minimum number of edges) between the vertices and . An optimal -coloring for is one minimizing the largest used integer over all such colorings. This coloring problem has relevant application in channel assignment for interference avoidance in wireless networks, where channels (i.e. colors) assigned to interfering stations (i.e. vertices) at distance must be at least apart, while the same channel can be reused only at stations whose distance is larger than . This paper presents efficient algorithms for finding optimal -colorings of trees and interval graphs as well as optimal -colorings of complete binary trees. Moreover, efficient algorithms are also provided for finding approximate -colorings of trees and interval graphs as well as approximate -colorings of unit interval graphs
An Exact Algorithm for the Generalized List -Coloring Problem
The generalized list -coloring is a common generalization of many graph
coloring models, including classical coloring, -labeling, channel
assignment and -coloring. Every vertex from the input graph has a list of
permitted labels. Moreover, every edge has a set of forbidden differences. We
ask for such a labeling of vertices of the input graph with natural numbers, in
which every vertex gets a label from its list of permitted labels and the
difference of labels of the endpoints of each edge does not belong to the set
of forbidden differences of this edge. In this paper we present an exact
algorithm solving this problem, running in time ,
where is the maximum forbidden difference over all edges of the input
graph and is the number of its vertices. Moreover, we show how to improve
this bound if the input graph has some special structure, e.g. a bounded
maximum degree, no big induced stars or a perfect matching
Learning-Based Constraint Satisfaction With Sensing Restrictions
In this paper we consider graph-coloring problems, an important subset of
general constraint satisfaction problems that arise in wireless resource
allocation. We constructively establish the existence of fully decentralized
learning-based algorithms that are able to find a proper coloring even in the
presence of strong sensing restrictions, in particular sensing asymmetry of the
type encountered when hidden terminals are present. Our main analytic
contribution is to establish sufficient conditions on the sensing behaviour to
ensure that the solvers find satisfying assignments with probability one. These
conditions take the form of connectivity requirements on the induced sensing
graph. These requirements are mild, and we demonstrate that they are commonly
satisfied in wireless allocation tasks. We argue that our results are of
considerable practical importance in view of the prevalence of both
communication and sensing restrictions in wireless resource allocation
problems. The class of algorithms analysed here requires no message-passing
whatsoever between wireless devices, and we show that they continue to perform
well even when devices are only able to carry out constrained sensing of the
surrounding radio environment
The Radio Number of Grid Graphs
The radio number problem uses a graph-theoretical model to simulate optimal
frequency assignments on wireless networks. A radio labeling of a connected
graph is a function such that for every pair
of vertices , we have where denotes the diameter of and
the distance between vertices and . Let be the
difference between the greatest label and least label assigned to . Then,
the \textit{radio number} of a graph is defined as the minimum
value of over all radio labelings of . So far, there have
been few results on the radio number of the grid graph: In 2009 Calles and
Gomez gave an upper and lower bound for square grids, and in 2008 Flores and
Lewis were unable to completely determine the radio number of the ladder graph
(a 2 by grid). In this paper, we completely determine the radio number of
the grid graph for , characterizing three subcases of the
problem and providing a closed-form solution to each. These results have
implications in the optimization of radio frequency assignment in wireless
networks such as cell towers and environmental sensors.Comment: 17 pages, 7 figure
Tight lower bound for the channel assignment problem
We study the complexity of the Channel Assignment problem. A major open
problem asks whether Channel Assignment admits an -time algorithm, for
a constant independent of the weights on the edges. We answer this question
in the negative i.e. we show that there is no -time algorithm
solving Channel Assignment unless the Exponential Time Hypothesis fails. Note
that the currently best known algorithm works in time so our lower bound is tight
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