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The Maximum Clique Problem
, 1999
"... Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computation ..."
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Cited by 150 (20 self)
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Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .
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 .
Exponential speedup of fixed parameter algorithms on K_3,3minorfree or K_5minorfree graphs
 in The 13th Anual International Symposium on Algorithms and Computation—ISAAC 2002
, 2002
"... We present a fixed parameter algorithm that constructively solves the kdominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ 34k n O(1)). In fact, we present our algorithm for any Hminorfree graph where H is a singlecrossing graph (can be drawn on the plane ..."
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Cited by 12 (5 self)
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We present a fixed parameter algorithm that constructively solves the kdominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ 34k n O(1)). In fact, we present our algorithm for any Hminorfree graph where H is a singlecrossing graph (can be drawn on the plane with at most one crossing) and obtain the algorithm for K3,3(K5)minorfree graphs as a special case. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, cliquetransversal set, kernels in digraphs, feedback vertex set and a series of vertex removal problems. Our work generalizes and extends the recent result of exponential speedup in designing fixedparameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.
Exponential Speedup of FixedParameter Algorithms for Classes of Graphs Excluding SingleCrossing Graphs as Minors
, 2002
"... We present a fixedparameter algorithm that constructively solves the kdominating set problem on any class of graphs excluding a singlecrossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in O(4 9.55 √ k n O(1) ) time. Examples of such graph classes are the ..."
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Cited by 5 (4 self)
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We present a fixedparameter algorithm that constructively solves the kdominating set problem on any class of graphs excluding a singlecrossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in O(4 9.55 √ k n O(1) ) time. Examples of such graph classes are the K3,3minorfree graphs and the K5minorfree graphs. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, cliquetransversal set, kernels in digraphs, feedback vertex set, and a collection of vertexremoval problems. Our work generalizes and extends the recent results of exponential speedup in designing fixedparameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.
The Bidimensionality Theory and Its . . .
, 2005
"... Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixedparameter algorithms and approximation algorithms for NPhard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that ..."
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Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixedparameter algorithms and approximation algorithms for NPhard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k × k grid graph (and similar graphs) grows with k, typically as Ω(k 2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval parameters, dominating set, edge dominating set, rdominating set, connected dominating set, connected edge dominating set, connected rdominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many
CliqueTransversal Sets in Cubic Graphs
, 2007
"... A cliquetransversal set S of a graph G is a set of vertices of G such that S meets all cliques of G. The cliquetransversal number, denoted τc(G), is the minimum cardinality of a cliquetransversal set in G. In this paper we present an upper bound and a lower bound on τc(G) for cubic graphs, and ch ..."
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A cliquetransversal set S of a graph G is a set of vertices of G such that S meets all cliques of G. The cliquetransversal number, denoted τc(G), is the minimum cardinality of a cliquetransversal set in G. In this paper we present an upper bound and a lower bound on τc(G) for cubic graphs, and characterize the extremal cubic graphs achieving the lower bound. In addition, we present a sharp upper bound on τc(G) for clawfree cubic graphs.