Results 1  10
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25
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 33 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
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.
Applications of the Matroid Parity Problem to Approximating Steiner Trees
, 1998
"... The Steiner tree problem in graphs requires to find a minimum size connected subgraph containing a given subset of nodes (given points). In this paper we consider this problem in three classes of graphs: where the maximum path distance is 2, where given points form a dominating set and where the giv ..."
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Cited by 8 (2 self)
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The Steiner tree problem in graphs requires to find a minimum size connected subgraph containing a given subset of nodes (given points). In this paper we consider this problem in three classes of graphs: where the maximum path distance is 2, where given points form a dominating set and where the given points form a vertex cover. As all these problems are MAXSNP hard, the issue is what approximation can be obtained in polynomial time. In the first case we obtain an approximation ratio (of the achieved size over the minimal one) 5 4 + 3 80 = 1:2875, in the second case we achieve 4/3, and in the last case we achieve 8=7 \Gamma 1=160. 1 Introduction One of the strongest results in matroid theory is the polynomial time solution of the parity problem in linear matroids. Briefly stated, we have a collection of 2 \Theta n matrices, and our goal is to find a maximum size subcollection such that they can be stacked into a single matrix of maximum rank. This problem was solved by Lov'asz [4]...
Applications of the Linear Matroid Parity Algorithm to Approximating Steiner Trees
, 2006
"... The Steiner tree problem in unweighted graphs requires to find a minimum size connected subgraph containing a given subset of nodes (terminals). In this paper we investigate applications of the linear matroid parity algorithm to the Steiner tree problem for two classes of graphs: where the terminals ..."
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Cited by 4 (0 self)
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The Steiner tree problem in unweighted graphs requires to find a minimum size connected subgraph containing a given subset of nodes (terminals). In this paper we investigate applications of the linear matroid parity algorithm to the Steiner tree problem for two classes of graphs: where the terminals form a vertex cover and where terminals form a dominating set. As all these problems are MAXSNPhard, the issue is what approximation can be obtained in polynomial time. The previously best approximation ratio for the first class of graphs (also known as unweighted quasibipartite graphs) is ≈ 1.217 (Gröpl et al. [4]) is reduced in this paper to 8/7 − 1/160 ≈ 1.137. For the case of graphs where terminals form a dominating set, an approximation ratio of 4/3 is achieved.
A linear algorithm for finding a maximal planar subgraph
 SIAM J. Disc. Math
, 2006
"... Abstract. We construct an optimal lineartime algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a nonplanar graph. Our solution is based on a fast data structure for incremental planarity testing ..."
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Cited by 4 (0 self)
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Abstract. We construct an optimal lineartime algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a nonplanar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new optimal planarity testing algorithm. Key words. Planar graphs, planarity testing, incremental algorithms, graph planarization, data structures, triconnectivity. AMS subject classifications. 05C10, 05C85, 68R10, 68Q25, 68W40 1. Introduction. Agraphisplanar
Algebraic Algorithms for Linear Matroid Parity Problems
"... We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the ma ..."
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Cited by 3 (1 self)
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We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the matrix multiplication exponent. This improves the O(mrω)time algorithm by Gabow and Stallmann, and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O(mr2) which does not need fast matrix multiplication. We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader’s disjoint Spath problem, we present an O(nω)time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably, and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O(n4)time randomized algorithm where n is the number of vertices, and an O(n3)time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω(n4) and Ω(n3) respectively. The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader’s disjoint Spath are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.
Parallel Algorithms for Maximal Linear Forests
 The Transactions of the IEICE
, 1997
"... . The maximal linear forest problem is to find, given a graph G = (V; E), a maximal subset of V that induces a linear forest. Three parallel algorithms for this problem are presented. The first one is randomized and runs in O(log n) expected time using n 2 processors on a CRCW PRAM. The second one ..."
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Cited by 2 (2 self)
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. The maximal linear forest problem is to find, given a graph G = (V; E), a maximal subset of V that induces a linear forest. Three parallel algorithms for this problem are presented. The first one is randomized and runs in O(log n) expected time using n 2 processors on a CRCW PRAM. The second one is deterministic and runs in O(log 2 n) time using n 4 processors on an EREW PRAM. The last one is deterministic and runs in O(log 5 n) time using n 3 processors on an EREW PRAM. The results put the problem in the class NC. Keywords: parallel algorithms, randomized parallel algorithms, graph algorithms, linear forests, maximal matchings, maximal independent sets. 1 Introduction Since Karp and Wigderson showed that the maximal independent set (MIS) problem is in the class NC [11], much work has been devoted to the study of parallel complexity of maximality problems. A typical maximality problem on graphs is to find either a maximal vertexinduced subgraph (MVIS) or a maximal edge...
Apptopinv  user’s guide
, 2003
"... The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NPcomplete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on HopcroftTarjan planarity testing algorit ..."
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Cited by 1 (1 self)
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The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NPcomplete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on HopcroftTarjan planarity testing algorithm, the spanningtree heuristic and various algorithms based on the cactustree heuristic. Apptopinv contains also a simulated annealing algorithm that can be used to improve the solutions obtained from other heuristics. Most of the heuristics have also a greedy version. We have implemented graph generators for complete graphs, complete kpartite graphs, complete hypercubes, random graphs, random maximum planar and outerplanar graphs and random regular graphs. Apptopinv supports three different graph file formats. Apptopinv is written in C++ programming language for Linuxplatform and GCC 2.95.3 compiler. To compile the program, a commercial LEDA algorithm
Finding Large Planar Subgraphs and Large Subgraphs of a Given Genus
 Proc. 2nd International Computing and Combinatorics Conference
, 1996
"... . We consider the MAXIMUM PLANAR SUBGRAPH problem  given a graph G, find a largest planar subgraph of G. This problem has applications in circuit layout, facility layout, and graph drawing. We improve to 4/9 the best known approximation ratio for the MAXIMUM PLANAR SUBGRAPH problem. We also conside ..."
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Cited by 1 (1 self)
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. We consider the MAXIMUM PLANAR SUBGRAPH problem  given a graph G, find a largest planar subgraph of G. This problem has applications in circuit layout, facility layout, and graph drawing. We improve to 4/9 the best known approximation ratio for the MAXIMUM PLANAR SUBGRAPH problem. We also consider a generalization of the previous problem, the MAXIMUM GENUS D SUBGRAPH problem  given a connected graph G, find a maximum subgraph of G of genus at most D. For the latter problem, we present a simple algorithm whose approximation ratio is 1/4. 1 Introduction The MAXIMUM PLANAR SUBGRAPH problem is: given a graph G, find a subgraph of G of maximum size, where size is the number of edges. This problem has applications in circuit layout, facility layout, and graph drawing [F92, TDB88]. MAXIMUM PLANAR SUBGRAPH is known to be NPComplete [LG77]. Therefore we are looking for polynomialtime approximation algorithms. For a graph G, let Opt(G) be the maximum size of a planar subgraph of G. Given ...
Optimal NodeDegree Bounds for the Complexity of Nonplanarity Parameters
 IN PROC. TENTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS, SODA'99
, 1999
"... We prove that both the NPcompleteness of the nonplanar deletion decision problem and the Max SNPhardness of the nonplanar deletion problem remain true even for cubic graphs. We prove that the class of graphs with splitting number less than or equal to a fixed k is minor closed, which implies the e ..."
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Cited by 1 (1 self)
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We prove that both the NPcompleteness of the nonplanar deletion decision problem and the Max SNPhardness of the nonplanar deletion problem remain true even for cubic graphs. We prove that the class of graphs with splitting number less than or equal to a fixed k is minor closed, which implies the existence of a corresponding polynomialtime recognition algorithm.