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A Polyhedral Approach to Planar Augmentation and Related Problems
, 1995
"... . Given a planar graph G, the planar (biconnectivity) augmentation problem is to add the minimum number of edges to G such that the resulting graph is still planar and biconnected. Given a nonplanar and biconnected graph, the maximum planar biconnected subgraph problem consists of removing the minim ..."
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Cited by 15 (1 self)
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. Given a planar graph G, the planar (biconnectivity) augmentation problem is to add the minimum number of edges to G such that the resulting graph is still planar and biconnected. Given a nonplanar and biconnected graph, the maximum planar biconnected subgraph problem consists of removing the minimum number of edges so that planarity is achieved and biconnectivity is maintained. Both problems are important in Automatic Graph Drawing. In [JM95], the minimum planarizing k augmentation problem has been introduced, that links the planarization step and the augmentation step together. Here, we are given a graph which is not necessarily planar and not necessarily kconnected, and we want to delete some set of edges D and to add some set of edges A such that jDj + jAj is minimized and the resulting graph is planar, kconnected and spanning. For all three problems, we have given a polyhedral formulation by defining three different linear objective functions over the same polytope, namely ...
A New Approximation Algorithm for Finding Heavy Planar Subgraphs
 ALGORITHMICA
, 1997
"... We provide the first nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NPHard problem of finding a heaviest planar subgraph in an edgeweighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM ..."
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Cited by 9 (2 self)
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We provide the first nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NPHard problem of finding a heaviest planar subgraph in an edgeweighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH had performance ratio exceeding 1=3, which is obtained by any algorithm that produces a maximum weight spanning tree in G. Based on the BermanRamaiyer Steiner tree algorithm, the new algorithm has performance ratio at least 1/3 + 1/72. We also show that if G is complete and its edge weights satisfy the triangle inequality, then the performance ratio is at least 3/8. Furthermore, we derive the first nontrivial performance ratio (7/12 instead of 1/2) for the NPHard MAXIMUM WEIGHT OUTERPLANAR SUBGRAPH problem.
Computing Orthogonal Drawings in a Variable Embedding Setting
 In [57
, 1998
"... This paper addresses the classical graph drawing problem of designing an algorithm that computes an orthogonal representation with the minimum number of bends. The algorithm receives as input a 4planar graph with a given ordering of the edges around the vertices and is allowed to change such orderi ..."
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Cited by 5 (1 self)
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This paper addresses the classical graph drawing problem of designing an algorithm that computes an orthogonal representation with the minimum number of bends. The algorithm receives as input a 4planar graph with a given ordering of the edges around the vertices and is allowed to change such ordering to reach the optimum. While the general problem has been shown to be NP complete [10], polynomial time algorithms have been devised for graphs whose vertex degree is at most three [5]. We show the first algorithm whose time complexity is exponential only in the number of vertices of degree four of the input graph. This settles a problem left as open in [6]. Our algorithm is further extended to handle graphs with vertices of degree higher than four. The analysis of the algorithm is supported by several experiments on the structure of a large set of input graphs. 1 Introduction and Overview Graph drawing is concerned with the design of methods for the automatic display of graphs so as to ...
Graph Drawing Algorithm Engineering with AGD
, 2000
"... We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basic ..."
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Cited by 3 (2 self)
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We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basically the same functionality. This enables us to exchange components and experiment with various algorithms and implementations of the same type. We give examples for algorithm engineering with AGD for drawing general nonhierarchical graphs and hierarchical graphs.
A Polynomial Time Randomized Parallel Approximation Algorithm for Finding Heavy Planar Subgraphs
, 2006
"... We provide an approximation algorithm for the Maximum Weight Planar Subgraph problem, the NPhard problem of finding a heaviest planar subgraph in an edgeweighted graph G. In the general case our algorithm has performance ratio at least 1/3 + 1/72 matching the best algorithm known so far, though in ..."
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We provide an approximation algorithm for the Maximum Weight Planar Subgraph problem, the NPhard problem of finding a heaviest planar subgraph in an edgeweighted graph G. In the general case our algorithm has performance ratio at least 1/3 + 1/72 matching the best algorithm known so far, though in several special cases we prove stronger results. In particular, we obtain performance ratio 2/3 (instead of 7/12) for the NPhard Maximum Weight Outerplanar Subgraph problem meeting the performance ratio of the best algorithm for the unweighted case. When the maximum weight planar subgraph is one of several special types of Hamiltonian graphs, we show performance ratios at least 2/5 and 4/9 (instead of 1/3 + 1/72), and 1/2 (instead of 4/9) for the unweighted case.
A GRASP FOR GRAPH PLANARIZATION ∗
"... Abstract. A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. In thispaper, we describe aGRASP forthe graph planarization problem, extending the heuristic of Goldschmidt and Takvorian [Networks, v. 24, pp. 69–73, 1994]. We review basic concepts of ..."
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Abstract. A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. In thispaper, we describe aGRASP forthe graph planarization problem, extending the heuristic of Goldschmidt and Takvorian [Networks, v. 24, pp. 69–73, 1994]. We review basic concepts of GRASP: construction and local search algorithms. The implementation of GRASP for graph planarization is described in detail. Computational experience on a large set of standard test problems is presented. On almost all test problems considered, the new heuristic either matches or finds a better solution than previously described graph planarization heuristics. In several cases, previously unknown optimal solutions are found. Key words. Combinatorial optimization, graph planarization, local search, GRASP, graph drawing, probabilistic algorithm, computer implementation 1. Introduction. A