![]() ![]() In: 2006 Pervasive Computing and Communications Workshops, p. 1.Barrett, C.L., Istrate, G., Anil Kumar, V.S., Marathe, M.V., Thite, S., Thulasidasan, S.: Strong edge coloring for channel assignment in wireless radio networks.This suggests several conjecties similar to the perfect graph conjecture. Also: for every optimal coloring there exists a path which meets each color class in only one point. In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable setS there exists a partition of the vertex set into paths which meet the stable set in only one point. of vertices in a path is at least equal to the chromatic number here again, we do not know if there exists an optimal coloring (S Later, Gallai proved that in a directed graph, the maximum number. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ1, μ2. Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. We discuss applications to important real world problems and identify areas for further work. ![]() Using these equivalences and recent results concerning the complexity of graph coloring, we classify many frequency assignment problems according to the "execution time efficiency" of algorithms that may be devised for their solution. We introduce two generalizations of chromatic number and show that many frequency assignment problems are equivalent to generalized graph coloring problems. A restricted class of graphs, called disk graphs, plays a central role in frequency-distance constrained problems. The frequency constrained approach should be avoided if distance separation is employed to mitigate interference. We model assignment problems as both frequency-distance constrained and frequency constrained optimization problems. This new approach is potentially more desirable than the traditional one. In this paper we introduce the minimum-order approach to frequency assignment and present a theory which relates this approach to the traditional one. We also consider related graph invariants, and the more general channel assignment model when assignments must satisfy "frequency-distance" constraints. We show that, as n 3 ,i n probability this ratio tends to 1 in the "sparse" case (when d d(n) is such that the average degree grows more slowly than ln n) and tends to 23/ 1.103 in the "dense" case (when the average degree grows faster than ln n). We consider the first n random points, and we are interested in particular in the behavior of the ratio of the chromatic number to the clique number. Thus we wish to color the nodes of a corresponding scaled unit disk graph. In the most basic version of the model, we assume that there is a threshold d such that, in order to avoid interference, points within distance less than d must be assigned distinct channels. In the model for random radio channel assignment considered here, points corre- sponding to transmitters are thrown down independently at random in the plane, and we must assign a radio channel to each point but avoid interference. (2) Finding a maximum clique in an input graph given in general form is NP-hard. The existence of this algorithm is to be reconciled with the apparent contradiction posed by the facts: (1) Recognizing whether an input graph given in general form is a unit disk graph is NP-hard in fact, it is not even known to be in NP. We give a polynomial time robust algorithm for the maximum clique problem in unit disk graphs, i.e., given an input graph G in general form, the output is either a maximum clique for G or a certificate that G is not a unit disk graph. An argument can be made that this hardness result is more meaningful than the trivial polynomial time promise algorithm. We show perhaps the surprising result that robustly finding a maximum independent set in a well-covered graph (i.e., a graph in which every maximal independent set is of the same size) is NP-hard. There exist problems that have a polynomial time promise solution, while being NP-hard if required to be robust. This is to be contrasted with the “promise” version of solving problems on restricted domains, in which there is a guarantee that the input is in the class, and an algorithm to “solve” the problem need not function correctly or even terminate if this guarantee is not met. Under this definition, an algorithm is required to be “robust,” i.e., it must produce correct output regardless of whether the input actually belongs to the restricted domain or not. We introduce a new definition of efficient algorithms for restricted domains. ![]()
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