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A spectral technique for coloring random 3colorable graphs
 SIAM Journal on Computing
, 1994
"... Abstract. Let G3n,p,3 be a random 3colorable 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 des ..."
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Cited by 79 (5 self)
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Abstract. Let G3n,p,3 be a random 3colorable 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 polynomialtime algorithm that finds a proper 3coloring of G3n,p,3 with high probability, whenever p ≥ 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 ≥ polylog(n)/n [J. Algorithms, 19 (1995), pp. 204–234]. The algorithm can be extended to produce optimal kcolorings of random kcolorable 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.
Approximation Results for the Optimum Cost Chromatic Partition Problem
 J. Algorithms
"... . In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove several approximation ..."
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Cited by 25 (0 self)
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. In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove several approximation results for the OCCP problem restricted to bipartite, chordal, comparability, interval, permutation, split and unimodular graphs. We prove that there exists no polynomial approximation algorithm with ratio O(jV j 0:5 ) for the OCCP problem restricted to bipartite and interval graphs, unless P = NP . Furthermore, we propose approximation algorithms with ratio O(jV j 0:5 ) for bipartite, interval and unimodular graphs. Finally, we prove that there exists no polynomial approximation algorithm with ratio O(jV j 1 ) for the OCCP problem restricted to split, chordal, permutation and comparability graphs, unless P = NP .
On the complexity of the Maximum Cut problem
 Nordic Journal of Computing
, 1991
"... The complexity of the simple maxcut problem is investigated for several special classes of graphs. It is shown that this problem is NPcomplete when restricted to one of the following classes: chordal graphs, undirected path graphs, split graphs, tripartite graphs, and graphs that are the complement ..."
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Cited by 14 (4 self)
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The complexity of the simple maxcut problem is investigated for several special classes of graphs. It is shown that this problem is NPcomplete when restricted to one of the following classes: chordal graphs, undirected path graphs, split graphs, tripartite graphs, and graphs that are the complement of a bipartite graph. The problem can be solved in polynomial time, when restricted to graphs with bounded treewidth, or cographs. We also give large classes of graphs that can be seen as generalizations of classes of graphs with bounded treewidth and of the class of the cographs, and allow polynomial time algorithms for the simple max cut problem. 1 Introduction One of the best known combinatorial graph problems is the max cut problem. In this problem, we have a weighted, undirected graph G = (V; E) and we look for a partition of the vertices of G into two disjoint sets, such that the total weight of the edges that go from one set to the other is as large as possible. In the simple max cu...
A Search Space “Cartography” for Guiding Graph Coloring Heuristics
 International Conferences
"... We present a search space analysis and its application in improving local search algorithms for the graph coloring problem. Using a classical distance measure between colorings, we introduce the following clustering hypothesis: the high quality solutions are not randomly scattered in the search spac ..."
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Cited by 9 (6 self)
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We present a search space analysis and its application in improving local search algorithms for the graph coloring problem. Using a classical distance measure between colorings, we introduce the following clustering hypothesis: the high quality solutions are not randomly scattered in the search space, but rather grouped in clusters within spheres of specific diameter. We first provide intuitive evidence for this hypothesis by presenting a projection of a large set of local minima in the 3D space. An experimental confirmation is also presented: we introduce two algorithms that exploit the hypothesis by guiding an underlying Tabu Search (TS) process. The first algorithm (TSDiv) uses a learning process to guide the basic TS process toward asyetunvisited spheres. The second algorithm (TSInt) makes deep investigations within a bounded region by organizing it as a treelike structure of connected spheres. We experimentally demonstrate that if such a region contains a global optimum, TSInt does not fail in eventually finding it. This pair of algorithms significantly outperforms the underlying basic TS algorithm; it can even improve some of the bestknown solutions ever reported in the literature (e.g. for dsjc1000.9). Key words: graph coloring, local optima distribution, search by learning. 1
Restrictions of Graph Partition Problems. Part I
 PART I. THEORETICAL COMPUTER SCIENCE
, 1991
"... In this paper, the problems to partition a given graph into k independent sets or cliques of bounded size k' are analysed for several classes of graphs. We ..."
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Cited by 4 (1 self)
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In this paper, the problems to partition a given graph into k independent sets or cliques of bounded size k' are analysed for several classes of graphs. We
Designing heterogeneous distributed GAs by efficiently selfadapting the migration period
, 2011
"... Abstract This paper investigates a new heterogeneous method that dynamically sets the migration period of a distributed Genetic Algorithm (dGA). Each island GA of this multipopulation technique selfadapts the period for exchanging information with the other islands regarding the local evolution pro ..."
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Cited by 2 (0 self)
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Abstract This paper investigates a new heterogeneous method that dynamically sets the migration period of a distributed Genetic Algorithm (dGA). Each island GA of this multipopulation technique selfadapts the period for exchanging information with the other islands regarding the local evolution process. Thus, the different islands can develop different migration settings behaving like a heterogeneous dGA. The proposed algorithm is tested on a large set of instances of the MaxCut problem, and it can be easily applied to other optimization problems. The results of this heterogeneous dGA are competitive with the best existing algorithms, with the added advantage of avoiding timeconsuming preliminary tests for tuning the algorithm.
An Approximation Algorithm for the License and Shift Class Design Problem
, 1994
"... In this paper a generalization of the license and shift class design problem is considered. Let J be a set of jobs which have to be carried out in specied time intervals, let E be a set of dierent engineer licenses and Z be a set of shifts. Given a price P (e; z) 2 Q0 + for engineers with licen ..."
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Cited by 1 (0 self)
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In this paper a generalization of the license and shift class design problem is considered. Let J be a set of jobs which have to be carried out in specied time intervals, let E be a set of dierent engineer licenses and Z be a set of shifts. Given a price P (e; z) 2 Q0 + for engineers with license e 2 E assigned to shift z 2 Z, the problem is to nd numbers of engineers to carry out all jobs with minimum cost. An approximation algorithm with time complexity O(jEj jZj jJ j 2 ) and approximation bound O(log(jJ j)) is proposed for this problem. Keywords: Design planning, scheduling, graphs, heuristics. 1 Introduction In this paper we consider a general interval scheduling problem where each job is classied by a xed start and nishing time. An application is given by jobs of aircraft inspections at an airport between the arrival and departure times [9, 10, 11]. Depending on the type of the aircraft such as B747 and DC10, each job is assigned an aircraft type. This impl...
Short Length Versions of Menger's Theorem (Extended Abstract)
"... Consider a simple nvertex undirected graph and assume there are ^ edgedisjoint paths between two vertices u and v. We prove the following two results: ffl There are ^ edgedisjoint paths between u and v, the average length of which is O(n=p^)ffl If all vertices have degree at least ^, there are ..."
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Consider a simple nvertex undirected graph and assume there are ^ edgedisjoint paths between two vertices u and v. We prove the following two results: ffl There are ^ edgedisjoint paths between u and v, the average length of which is O(n=p^)ffl If all vertices have degree at least ^, there are ^ edgedisjoint paths between u and v, each of which has length O(n=^). These bounds are best possible. For directed graphs, the first result still holds but not the second. Some of the paths can be at least \Omega (n) long. We also describe how to use a minimum cost flow algorithm to find the paths implied by the above results in time O(^m). In a ^ edgeconnected graph, we define the concept of bistance (or bulk distance). The bistance between u and v is the minimum over all ^ edgedisjoint paths between u and v of the maximum path length. We prove that bistance forms a metric. We give NPhardness results on computing bistances in two cases. The third remaining case is open but we give evidence to its difficulty.
Solving Large Instances of the Quadratic Cost Partition Problem on Dense Graphs by DataCorrecting Algorithms: a Computational Study
, 1999
"... The DataCorrecting Algorithm (DCA) corrects the data of a hard problem instance in such a way that we obtain an instance of a well solvable special case. For a given prescribed accuracy of the solution, the DCA uses a branch and bound scheme to make sure that the solution of the corrected instance ..."
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The DataCorrecting Algorithm (DCA) corrects the data of a hard problem instance in such a way that we obtain an instance of a well solvable special case. For a given prescribed accuracy of the solution, the DCA uses a branch and bound scheme to make sure that the solution of the corrected instance satisfies this given prescribed accuracy. We describe the “hardness ” of randomly generated instances of the Quadratic Cost Partition Problem (QCP) by the density of the corresponding graphs as well as by the cardinality of an optimal solution of the QCP. We study the behaviour of the DCAs through the number of search levels of the so called Preliminary Preservation Algorithm and average computational times. We report average computational times of the DCA for instances of dense graphs with the number of vertices up to 500 which can be solved on a standard PC within 10 minutes.