A general approximation technique for constrained forest problems

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A general approximation technique for constrained forest problems (1995)

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by Michel X. Goemans , David P. Williamson

Venue:	SIAM J. COMPUT.
Citations:	414 - 21 self

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@MISC{Goemans95ageneral,
    author = {Michel X. Goemans and David P. Williamson},
    title = {A general approximation technique for constrained forest problems},
    year = {1995}
}

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Abstract

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, minimum-cost spanning tree, minimum-weight 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 2-approximation algorithm for the minimum-weight 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 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the nonfixed point-to-point connection problem, the exact path partitioning problem, and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413-420].

Keyphrases

general approximation technique 2-approximation algorithm exact path partitioning problem certain requirement polynomial exact algorithm similar result minimum-weight perfect large class 2-matching problem nonfixed point-to-point connection problem dense graph triangle inequality minimum-cost spanning tree running time complex location-design problem minimum-weight perfect matching first approximation algorithm steiner tree problem many np-complete problem prize-collecting traveling salesman approximation algorithm best-known performance guarantee graph problem many basic combinatorial optimization problem minimum cost

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