We introduce and explore a family of vertex-coloring problems, which, surprisingly enough, have not been considered before despite stemming from the problem of Wi-Fi channel assignment. Given a spectrum of colors, endowed with a matrix of interferences between each pair of colors, the Threshold Spectrum Coloring problem fixes the number of colors available and aims to minimize the interference threshold, i.e., the maximum of the interferences at the vertices. Conversely, the Chromatic Spectrum Coloring problem fixes a threshold and aims to minimize the number of colors for which respecting that threshold is possible. As the main theoretical results, we prove tight upper bounds for the solutions to each problem. Since both problems turn out to be NP-hard, we complete the scene with experimental results. We propose a DSATUR-based heuristic and study its performance to minimize the maximum vertex interference in Wi-Fi channel assignment, both for randomly-generated graphs and for a real-world scenario. Further, for all these graphs, we experimentally check the goodness of the theoretical bounds.
@cite_11 consider a weight function @math on the edges of the graph @math instead of a matrix @math of interferences between colors, defining the interference at a vertex to be the sum of weights of incident monochromatic edges. Although they bounds are similar to ours, using the same techniques, we further show tightness and follow a different, more detailed, scheme. Many other works impose conditions to the colors of the endpoints of any edge. Most of them can be framed into @cite_7 , where vertices at distance @math must get colors at distance @math . Particular instances include @cite_8 , where no two vertices at distance @math can have the same color, @cite_17 , where adjacent vertices must get colors @math apart and vertices at distance @math must get different colors, as well as @cite_13 , where adjacent vertices must get colors @math apart and vertices at distance @math must get colors @math apart.
@cite_11 @cite_7 @cite_8 @cite_13 @cite_17
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Let @math be a graph with a threshold function @math such that @math for every vertex @math of @math , where @math is the degree of @math in @math . Suppose we are given a target set @math . This paper considers the following repetitive process on @math . At time step @math the vertices of @math are colored black and the other vertices are colored white. After that, at each time step @math , the colors of white vertices (if any) are updated according to the following rule. All white vertices @math that have at least @math black neighbors at the time step @math are colored black, and the colors of the other vertices do not change. The process runs until no more white vertices can update colors from white to black. The following optimization problem is called Target Set Selection: Find a target set @math of smallest possible size such that all vertices in @math are black at the end of the process. Such an @math is called an optimal target set for @math under the th... We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph @math on @math vertices has a rainbow cycle on at least @math vertices, by showing that @math has a rainbow cycle on at least @math vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least @math different colors appear. For large @math , this is an improvement of the previous best known lower bound of @math of Andersen. Determining whether the vertices of a graph can be colored using @math different colors so that no two adjacent vertices receive the same color is a well-known NP-complete problem. Graph coloring is also of practical interest (for example, in estimating sparse Jacobians and in scheduling), and many heuristic algorithms have been developed. We present a heuristic algorithm based on the eigenvalue decomposition of the adjacency matrix of a graph. Eigenvectors point out bipartite-looking'''' subgraphs that are used to refine the coloring to a valid coloring. The algorithm optimally colors complete @math -partite graphs and certain other classes of graphs with regular structure. Let G=(V,E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of [SIAM J. Comput., 28 (1999), pp. 1167--1181], we describe an @math -time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm. Furthermore, for every even k>2, we describe an @math -time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an @math -time algorithm @math for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in @math time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every @math we have @math . We show that every unweighted graph on n vertices has a 2-emulator with @math edges and a 4-emulator with @math edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with @math edges and that such a 3-spanner can be built in @math time. We also describe an @math -time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3. Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p @math c n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p @math polylog(n) n [J. Algorithms, 19 (1995), pp. 204--234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c.