Edge‐maximal triangulated subgraphs and heuristics for the maximum clique problem

Xue, Jue

doi:10.1002/net.3230240208

Cited by 29 publications

(21 citation statements)

References 11 publications

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“…We should mention that this problem is solvable in time O( m), where is the highest degree in the input graph [3,36,83]. An interesting question is whether faster algorithms for it can be found.…”

Section: Discussionmentioning

confidence: 97%

Minimal triangulations of graphs: A survey

Heggernes

2006

Discrete Mathematics

155

110

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Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.

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Section: Discussionmentioning

confidence: 97%

Minimal triangulations of graphs: A survey

Heggernes

2006

Discrete Mathematics

155

110

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“…We denote by LBX (°LBP) the cost of the feasible solution of problem WNP obtained by means of the heuristic algorithm proposed by Xue (1994).…”

Section: Generation Of Graph G å (X E)mentioning

confidence: 99%

An Exact Algorithm for the Resource-Constrained Project Scheduling Problem Based on a New Mathematical Formulation

et al. 1998

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In this paper we consider the Project Scheduling Problem with resource constraints, where the objective is to minimize the project makespan. We present a new 0-1 linear programming formulation of the problem that requires an exponential number of variables, corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints. Different relaxations of the above formulation are used to derive new lower bounds, which dominate the value of the longest path on the precedence graph and are tighter than the bound proposed by Stinson et al. (1978). A tree search algorithm, based on the above formulation, that uses new lower bounds and dominance criteria is also presented. Computational results indicate that the exact algorithm can solve hard instances that cannot be solved by the best algorithms reported in the literature.Project Scheduling, Branch-and-Bound Methods, Networks/Graphs, Lower Bounds

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“…Some heuristics for ÿnding a large clique (see, e.g., [39]) aim to ÿnd a maximum chordal subgraph of the input graph, on which a maximum clique can be found in polynomial time.…”

Section: Motivationmentioning

confidence: 99%

Complexity Classification of Some Edge Modification Problems

Natanzon¹,

Shamir²,

Sharan³

1999

Graph-Theoretic Concepts in Computer Science

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In an edge modiÿcation problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NP-hardness of a variety of edge modiÿcation problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a ÿnite set of forbidden induced subgraphs. ?

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Edge‐maximal triangulated subgraphs and heuristics for the maximum clique problem

Cited by 29 publications

References 11 publications

Minimal triangulations of graphs: A survey

Minimal triangulations of graphs: A survey

An Exact Algorithm for the Resource-Constrained Project Scheduling Problem Based on a New Mathematical Formulation

Complexity Classification of Some Edge Modification Problems

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