Abstract
We introduce flow metrics as a relaxation of path metrics (i.e. linear orderings). They are defined by polynomial-sized linear programs and have interesting properties including spreading. We use them to obtain relaxations for several NP-hard linear ordering problems such as the minimum linear arrangement and minimum pathwidth. Our approach has the advantage of achieving the best-known approximation guarantees for these problems using the same relaxation and essentially the same rounding for all the problems and varying only the objective function from problem to problem. This is in contrast to the current state of the literature where each problem warrants either a new relaxation or a new rounding or both. We also characterize a natural projection of the relaxation.
Supported in part by CNPq grant 300083/99-8 and a Protem CNPq/NSF joint Grant.
Supported in part by NSF Career Award CCR-9875024.
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References
Y. Bartal, “Probabilistic Approximation of Metric Spaces and its Algorithmic Applications,” Proc. of the 37th Ann. IEEE Symp. on Foundations of Computer Science, 184–193, 1996.
A. Blum, G. Konjevod, R. Ravi and S. Vempala, “Semi-Definite Relaxation for Minimum Bandwidth and other Vertex-Ordering Problems,” Theoretical Computer Science, 235 (2000), 25–42. Preliminary version in Proc. 30th ACM Symposium on the Theory of Computing, Dallas, 1998.
H. L. Bodlaender, J. R. Gilbert, H. Hafsteinsson, T. Klok, “Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height,” Journal of Algorithms 18 (1995) 238–255.
M. M. Deza and M. Laurent, Geometry of cuts and metrics, Springer-Verlag, 1997.
J. Dunagan and S. Vempala, “On Euclidean embeddings and bandwidth minimization,” Proc. of the 5th Intl. Symp. on Randomization and Approximation techniques in Computer Science, 229–240, 2001.
G. Even. J. Naor, S. Rao and B. Schieber, “Divide-and-conquer approximation algorithms via spreading metrics,” Proceedings of the 35th Annual Conference on Foundations of Computer Science, 62–71, 1995.
M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, 1988.
M. Goemans and D. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, JACM, 42: 1115–1145, 1995.
T. Leighton and S. Rao. “An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms.” In Proc. of 28th FOCS, pp256–69, 1988.
S. Rao and A. Richa, “New Approximation Techniques for Some Ordering Problems,” Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 211–218, 1998.
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© 2002 Springer-Verlag Berlin Heidelberg
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Bornstein, C.F., Vempala, S. (2002). Flow Metrics. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_45
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DOI: https://doi.org/10.1007/3-540-45995-2_45
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