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A general approximation technique for constrained forest problems
 SIAM J. COMPUT.
, 1995
"... We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization proble ..."
Abstract

Cited by 358 (21 self)
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We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimumcost spanning tree, minimumweight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in O(n log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2approximation algorithm for the minimumweight perfect matching problem under the triangle inequality. Our running time of O(n log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n 3) time for dense graphs. A similar result is obtained for the 2matching problem and its variants. We also derive the first approximation algorithms for many NPcomplete problems, including the nonfixed pointtopoint connection problem, the exact path partitioning problem, and complex locationdesign problems. Moreover, for the prizecollecting traveling salesman or Steiner tree problems, we obtain 2approximation algorithms, therefore improving the previously bestknown performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413420].
THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
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Cited by 124 (7 self)
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The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
LiNIEAR‘I’IMIE APPROXIMATION ALGORITHMS FOR FINDING THE MUUMUM~WEIGHT PERFECT ~TCH~G ON A PLANE
, 1981
"... We consider the problem of determining the minimumweight perfect matching of n [even) points on a plane, i.e,, determining how to match the n points in pairs so as to rn~ ~ the sum of the distances between the matched points. This problem is of fundamental imFor%!nce in fmding the optimal sequence ..."
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We consider the problem of determining the minimumweight perfect matching of n [even) points on a plane, i.e,, determining how to match the n points in pairs so as to rn~ ~ the sum of the distances between the matched points. This problem is of fundamental imFor%!nce in fmding the optimal sequence of drawing edges of a coM~ted graph by a rne~h~i~ ~ plotter; as is easgy confirmed, the wasted plotterpen movement is minimized by fmding a minimumweight, perfect matching of the vertices of an odd degree, adding the edges between the matched pairs to the original graph and traver~g an Eulerian p&t on the extended graph, where the added edges are to be traversed with the pen off the paper 1131. The algorithm [2] which exactly solves this problem in o(n3) time seems to be too complicated from the practical point of view. Even approximation algorithms of 0(n2) or O(n log n) [4] wouid not be satisfactory for the application to plotters, since an Euferian path can be found i.. time linear in the number of edges, In fact, our Monte Carlo expe~ents for 1400 problems with 32 to 2048 points have shown that the strip algorithm of O(n log n) [4] requires 3 to 10 times as much ~omput~g time as our ~~e~t~e algorithms do. In this paper, lineartime approximation algorithms are proposed for the mat&ing problem on a unit square and the worstcase performance is ~~yzed ~eore~~~y, The quality of an approbate solution is measured by the ~~so~~~e cost of the matching, i.e., the sum of the distances between the paired points, and not by the ratio of the cost to that of the exact opts solution. As the distance, not only the L2distance (Euclidean distance) but also the L,distance (maximum norm) is considered, the latter being more appropriate when the big time of a mechanical plotter is in question. 2. Algorithm To be specific, we describe here ‘the serpentine
New Primal and Dual Matching Heuristics 1
, 1995
"... Abstract. We describe a new heuristic for constructing a minimumcost perfect matching designed for problems on complete graphs whose cost functions satisfy the triangle inequality (e.g., Euclidean problems): The running time for an n node problem is O(n log n) after a minimumcost spanning tree is ..."
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Abstract. We describe a new heuristic for constructing a minimumcost perfect matching designed for problems on complete graphs whose cost functions satisfy the triangle inequality (e.g., Euclidean problems): The running time for an n node problem is O(n log n) after a minimumcost spanning tree is constructed. We also describe a procedure which, added to Kruskal's algorithm, produces a lower bound on the size of any perfect matching. This bound is based on a dual problem which has the following geometric interpretation for Euclidean problems: Pack nonoverlapping disks centered at the nodes and moats surrounding odd sets of nodes so as to maximize the sum of the disk radii and moat widths.
Fast and Simple Algorithms for Weighted Perfect Matching
"... We present two fast and simple combinatorial approximation algorithms for constructing a minimumweighted perfect matching on complete graphs whose cost functions satisfy the triangle inequality. The first algorithm runs in O(n 2 log n) time and is at most a factor log n worse than an optimal soluti ..."
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We present two fast and simple combinatorial approximation algorithms for constructing a minimumweighted perfect matching on complete graphs whose cost functions satisfy the triangle inequality. The first algorithm runs in O(n 2 log n) time and is at most a factor log n worse than an optimal solution. In the second algorithm, the average time until a node is matched is O(n 2) and the approximation ratio is log 2 n. Key words: weighted matching, approximation algorithms 1
Parallel Algorithms in Graph Theory and Algebra
"... published in the European Association for Theoretical Computer Science Bulletin, Number 50, June 1993. ffl Presented at the Algorithms and Complexity II project workshop, held in Homburg, Germany, September 1993. ffl Presented at Workshop on Graphs '93. Published[15] in the proceedings: vi N ..."
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published in the European Association for Theoretical Computer Science Bulletin, Number 50, June 1993. ffl Presented at the Algorithms and Complexity II project workshop, held in Homburg, Germany, September 1993. ffl Presented at Workshop on Graphs '93. Published[15] in the proceedings: vi N.W. Holloway, S. Ravindran, and A.M. Gibbons. "Approximating minimumweight perfect matchings for complete graphs satisfying the triangle inequality", In GraphTheoretic Concepts in Computer Science, Jan van Leeuwen (editor), volume 790 of LLNCS, pages 1120. SpringerVerlag, 1993. Unifiable Algebras Joint work of N.W. Holloway and A.M. Gibbons. ffl Presented at the 10th annual meeting of the British Colloquium for Theoretical Computer Science, held in Bristol, UK, March 1994. Abstract to be published in the European Association for Theoretical Computer Science Bulletin, Number 55, February 1995. Studying Complexities of Genetic Algorithms Joint work of M. Amos, S. Ravindran, N.W. Holloway and A.M. Gibbons. ffl Presented at the 10th annual meeting of the British Colloquium for Theoretical Computer Science, held in Bristol, UK, March 1994. Abstract to be published in the European Association for Theoretical Computer Science Bulletin, Number 55, February 1995. vii Summary Chapter 1 describes the model of parallel computation used, and some standard algorithmic techniques that can be used to construct efficient parallel algorithms. Chapter 2 presents an NC approximation for finding a minimum weight perfect matching in complete graphs. This is the first approximation algorithm for this problem with a sublinear performance ratio. This algorithm is of interest as matching problems are on the boundary of what problems may be described as tractable for parallel computation....
Acquisitions and Acquisitions et Bibliographie Services services bibliographiques
, 1999
"... Our fi & N o m reldrenca ..."