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Antibandwidth of Complete kAry Trees
, 2006
"... The antibandwidth problem is to label vertices of a nvertex graph injectively by 1, 2, 3,... n, such that the minimum difference of labels of adjacent vertices is maximised. The problem is motivated by obnoxious facility location problem, radiocolouring, work and game scheduling and is dual to the ..."
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The antibandwidth problem is to label vertices of a nvertex graph injectively by 1, 2, 3,... n, such that the minimum difference of labels of adjacent vertices is maximised. The problem is motivated by obnoxious facility location problem, radiocolouring, work and game scheduling and is dual to the well known bandwidth problem. We prove exact results for the antibandwidth of complete kary trees, k even, and estimate the parameter for odd k up to the second order term. This extends previous results for complete binary trees.
A Note on Maximum Differential Coloring of Planar Graphs
"... We study the maximum differential coloring problem, where the vertices of an nvertex graph must be labeled with distinct numbers ranging from 1 to n, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is NPhard for general graphs [1 ..."
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We study the maximum differential coloring problem, where the vertices of an nvertex graph must be labeled with distinct numbers ranging from 1 to n, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is NPhard for general graphs [16], we consider planar graphs and subclasses thereof. We prove that the maximum differential coloring problem remains NPhard, even for planar graphs. We also present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the MillerPritikin labeling scheme [19] for forests is optimal for regular caterpillars and for spider graphs. 1.
An Improved Memetic Algorithm for the Antibandwidth Problem?
"... Abstract. This paper presents an Improved Memetic Algorithm (IMA) designed to compute nearoptimal solutions for the antibandwidth problem. It incorporates two distinguishing features: an efficient heuristic to generate a good quality initial population and a local search operator based on a Stoch ..."
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Abstract. This paper presents an Improved Memetic Algorithm (IMA) designed to compute nearoptimal solutions for the antibandwidth problem. It incorporates two distinguishing features: an efficient heuristic to generate a good quality initial population and a local search operator based on a Stochastic Hill Climbing algorithm. The most suitable combination of parameter values for IMA is determined by employing a tunning methodology based on Combinatorial Interaction Testing. The performance of the finetunned IMA algorithm is investigated through extensive experimentation over well known benchmarks and compared with an existing stateoftheart Memetic Algorithm, showing that IMA consistently improves the previous bestknown results.
Tabu Search for the Cyclic Bandwidth ProblemI
"... The Cyclic Bandwidth problem (CB) for graphs consists in labeling the vertices of a guest graph G by distinct vertices of a host cycle Cn (both of order n) in such a way that the maximum distance in the cycle between adjacent vertices in G is minimized. To the best of our knowledge, this is the firs ..."
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The Cyclic Bandwidth problem (CB) for graphs consists in labeling the vertices of a guest graph G by distinct vertices of a host cycle Cn (both of order n) in such a way that the maximum distance in the cycle between adjacent vertices in G is minimized. To the best of our knowledge, this is the first research work investigating the use of metaheuristic algorithms for solving this challenging combinatorial optimization problem in the case of general graphs. In this paper a new carefully devised Tabu Search algorithm, called TScb, for finding nearoptimal solutions for the CB problem is proposed. Different possibilities for its key components and input parameter values were carefully analyzed and tuned, in order to find the combination of them offering the best quality solutions to the problem at a reasonable computational effort. Extensive experimentation was carried out, using 113 standard benchmark instances, for assessing its performance with respect to a Simulated Annealing (SAcb) implementation. The experimental results show that there exists a statistically significant performance amelioration achieved by TScb with respect to SAcb in 90 out of 113 graphs (79.646%). It was also found that our TScb algorithm attains 56 optimal solutions and establishes new better upper bounds for the other 57 instances. Furthermore, these competitive results were obtained expending reasonable computational times. Key words: cyclic bandwidth problem, tabu search, bestknown bounds 1.
Maximum Differential Coloring of Caterpillars and Spiders
"... Abstract. We study the maximum differential coloring problem, where the vertices of an nvertex graph must be labeled with the numbers 1,..., n such that the minimum difference between the two labels of any adjacent vertices is maximized. As it is NPhard to find the optimal labels for general gra ..."
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Abstract. We study the maximum differential coloring problem, where the vertices of an nvertex graph must be labeled with the numbers 1,..., n such that the minimum difference between the two labels of any adjacent vertices is maximized. As it is NPhard to find the optimal labels for general graphs, we consider special subclasses: caterpillars, spiders, and extended stars. We first prove new upper bounds for maximum differential coloring for spiders and regular caterpillars. Using these new bounds, we prove that the MillerPritikin [18] labeling scheme for forests is optimal for regular caterpillars and for spider graphs. On the other hand, we give examples of general caterpillars where the MillerPritikin algorithm behaves very poorly. We present an alternative algorithm for general caterpillars which achieves reasonable results even in worst case. Finally we present new optimal labeling schemes for regular caterpillars and for subclasses of spider graphs. 1
Antibandwidth and Cyclic Antibandwidth of
"... The antibandwidth problem consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximized. The problem was originally introduced in [15] in connection with multiprocessor scheduling problems and can be al ..."
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The antibandwidth problem consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximized. The problem was originally introduced in [15] in connection with multiprocessor scheduling problems and can be also understood as a dual problem to the well known bandwidth problem, as a special radiocolouring problem or as a variant of obnoxious facility location problems. The antibandwidth problem is NPhard, there are a few classes of graphs with polynomial time complexities. Exact results for nontrivial graphs are very rare. Miller and Pritikin [18] showed tight bounds for 2dimensional meshes and hypercubes. We solve the antibandwidth problem precisely for two dimensional meshes, tori and estimate the antibandwidth value for hypercubes up to the third order term. The cyclic antibandwidth problem is to embed an nvertex graph into the cycle Cn, such that the minimum distance (measured in the cycle) of adjacent vertices is maximised. This is a natural extension Preprint submitted to Elsevier Science 27 October 2005 of the antibandwidth problem or a dual problem to the cyclic bandwidth problem. We start investigating this invariant for typical graphs and prove basic facts and exact results for the same product graphs as for the antibandwidth.
The Cyclic Antibandwidth Problem (Extended Abstract)
"... The cyclic antibandwidth problem is to embed an nvertex graph into the cycle Cn, such that the minimum distance (measured in the cycle) of adjacent vertices is maximised. This is a variant of obnoxious facility location problems or a dual problem to the cyclic bandwidth problem. The problem is NPh ..."
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The cyclic antibandwidth problem is to embed an nvertex graph into the cycle Cn, such that the minimum distance (measured in the cycle) of adjacent vertices is maximised. This is a variant of obnoxious facility location problems or a dual problem to the cyclic bandwidth problem. The problem is NPhard. In the paper we start investigating this invariant for typical graphs. We prove basic facts and exact results for meshes, tori and asymptotics for hypercubes. 1