Abstract
We introduce the notion of the width bounded geometric separator and develop the techniques for the existence of the width bounded separator in any d-dimensional Euclidean space. The separator is applied in obtaining \(2^{O(\sqrt{n})}\) time exact algorithms for a class of NP-complete geometric problems, whose previous algorithms take \(n^{O(\sqrt{n})}\) time[2][5][1]. One of those problems is the well known disk covering problem, which seeks to determine the minimal number of fixed size disks to cover n points on a plane[10]. They also include some NP-hard problems on disk graphs such as the maximum independent set problem, the vertex cover problem, and the minimum dominating set problem.
This research is supported by Louisiana Board of Regents fund under contract number LEQSF(2004-07)-RD-A-35.
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References
Agarwal, P., Overmars, M., Sharir, M.: Computing maximally separated sets in the plane and independent sets in the intersection graph of unit graph. In: Proceedings of 15th ACM-SIAM symposium on discrete mathematics algorithms, pp. 509–518. ACM-SIAM (2004)
Agarwal, P., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33, 201–226 (2002)
Alber, J., Fernau, H., Niedermeier, R.: Graph separators: a parameterized view. In: Proceedings of 7th Internal computing and combinatorics conference, pp. 318–327 (2001)
Alber, J., Fernau, H., Niedermeier, R.: Parameterized complexity: exponential speed-up for planar graph problems. In: Proceedings of 28st international colloquium on automata, languages and programming, pp. 261–272 (2001)
Alber, J., Fiala, J.: Geometric separation and exact solution for parameterized independent set problem on disk graphs. Journal of Algorithms 52(2), 134–151 (2004)
Alon, N., Seymour, P., Thomas, R.: Planar Separator. SIAM J. Discr. Math. 7(2), 184–193 (1990)
Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete mathematics 86, 165–177 (1990)
Djidjev, H.N.: On the problem of partitioning planar graphs. SIAM journal on discrete mathematics 3(2), 229–240 (1982)
Djidjev, H.N., Venkatesan, S.M.: Reduced constants for simple cycle graph separation. Acta informatica 34, 231–234 (1997)
Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Information processing letters 3(12), 133–137 (1981)
Fu, B., Wang, W.: A \(2^{O(n^{1-1/d}\log n)}\)-time algorithm for d-dimensional protein folding in the HP-model. In: Proceedings of 31st international colloquium on automata, languages and programming, pp. 630–644 (2004)
Fu, B., Chen, Z.: Sublinear-time algorithms for width-bounded geometric separator and their application to protein side-chain packing problem (submitted)
Gazit, H.: An improved algorithm for separating a planar graph, manuscript, USC (1986)
Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separation theorem for graphs of bounded genus. Journal of algorithm (5), 391–407 (1984)
Graham, R., Grötschel, M., Lovász, L.: Handbook of combinatorics, vol. I. MIT Press, Cambridge (1996)
Hales, T.C.: A computer verification of the Kepler conjecture. In: Proceedings of the ICM, Beijing, vol. 3, pp. 795–804 (2002)
Lichtenstein, D.: Planar formula and their uses. SIAM journal on computing 11(2), 329–343 (1982)
Lipton, R.J., Tarjan, R.: A separator theorem for planar graph. SIAM Journal on Applied Mathematics 36, 177–189 (1979)
Lipton, R.J., Tarjan, R.: Applications of a planar separator theorem. SIAM journal on computing 9(3), 615–627 (1980)
Meggido, N., Supowit, K.: On the complexity of some common geometric location problems. SIAM journal on computing 13, 1–29 (1984)
Miller, G.L., Teng, S.-H., Vavasis, S.A.: An unified geometric approach to graph separators. In: 32nd annual symposium on foundation of computer science, pp. 538–547. IEEE, Los Alamitos (1991)
Miller, G.L., Thurston, W.: Separators in two and three dimensions. In: 22nd Annual ACM symposium on theory of computing, pp. 300–309. ACM, New York (1990)
Ravi, S.S., Hunt III, H.B.: Application of the planar separator theorem to computing problems. Information processing letter 25(5), 317–322 (1987)
Smith, W.D., Wormald, N.C.: Application of geometric separator theorems. In: The 39th annual symposium on foundations of computer science, pp. 232–243 (1998)
Spielman, D.A., Teng, S.H.: Disk packings and planar separators. In: The 12th annual ACM symposium on computational geometry, pp. 349–358 (1996)
Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley-Interscience Publication, Chichester (1995)
Wong, D.W., Kuo, Y.S.: A study of two geometric location problems. Information processing letters 28, 281–286 (1988)
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Fu, B. (2006). Theory and Application of Width Bounded Geometric Separator. In: Durand, B., Thomas, W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11672142_22
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DOI: https://doi.org/10.1007/11672142_22
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