Given an edge-weighted graph G, the minimum maximal matching problem asks to find a minimum weight maximal matching. The problem is known to be NP-hard even if the graph is planar and unweighted. In this paper, we con...
详细信息
Given an edge-weighted graph G, the minimum maximal matching problem asks to find a minimum weight maximal matching. The problem is known to be NP-hard even if the graph is planar and unweighted. In this paper, we consider the problem in planar graphs. First, we prove a strong inapproximability for the problem in weighted planar graphs. Second, in contrast with the first result, we show that a polynomial time approximation scheme (PTAS) for the problem in unweighted planar graphs can be obtained by a divide-and-conquer method based on the planar separator theorem. For a given epsilon > 0, our scheme delivers in O(n log n + alphaepsilon(1/2) epsilon(-1)n) time a solution with size at most (1 + epsilon) times the optimal value, where n is the number of vertices in G and a is a constant number.
The local sequence alignment problem is the detection of similar subsequences in two given sequences of lengths n ≥ m. Unfortunately the common notion of local alignment suffers from some well-known anomalies which r...
详细信息
We study approximation bounds for the semidefinite programming (SDP) relaxation of quadratically constrained quadratic optimization: min f(0)(x) subject to f(k)(x) less than or equal to 0, k = 1,..., m, where f(k)(x) ...
详细信息
We study approximation bounds for the semidefinite programming (SDP) relaxation of quadratically constrained quadratic optimization: min f(0)(x) subject to f(k)(x) less than or equal to 0, k = 1,..., m, where f(k)(x) = x(T)A(k)x + (b(k))(T) x + c(k). In the special case of ellipsoid constraints with interior feasible solution at 0, we show that the SDP relaxation, coupled with a rank-1 decomposition result of Sturm and Zhang [ Math. Oper. Res., to appear], yields a feasible solution of the original problem with objective value at most (1 - gamma)(2)/(rootm + gamma)(2) times the optimal objective value, where gamma = rootmax(k) f(k)(0) + 1. For the single trust-region problem corresponding to m = 1, this yields an exact optimal solution. In the general case, we extend some bounds derived by Nesterov [Optim. Methods Softw., 9 (1998), pp. 141 - 160;working paper, CORE, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium, 1998], Ye [Math. Program., 84 (1999), pp. 219 - 226], and Nesterov, Wolkowicz, and Ye [in Handbook of Semidefinite Programming, H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, pp. 360 - 419] for the special case where A(k) is diagonal and b(k) = 0 for k = 1,..., m. We also discuss the generation of approximate solutions with high probability.
We show that the metric multilevel facility location problem is polynomial-time reducible within a factor of 3 to the metric uncapacitated facility location problem. This leads to a combinatorial 4.83-approximation al...
详细信息
We show that the metric multilevel facility location problem is polynomial-time reducible within a factor of 3 to the metric uncapacitated facility location problem. This leads to a combinatorial 4.83-approximation algorithm for the metric multilevel facility location problem and to a 9-approximation algorithm for a capacitated version of it. (C) 2002 Elsevier Science B.V. All rights reserved.
Semidefinite programming based approximation algorithms, such as the Goemans and Williamson approximation algorithm for the MAX CUT problem, are usually shown to have certain performance guarantees using local ratio t...
详细信息
Semidefinite programming based approximation algorithms, such as the Goemans and Williamson approximation algorithm for the MAX CUT problem, are usually shown to have certain performance guarantees using local ratio techniques. Are the bounds obtained in this way tight? This problem was considered before by Karloff [SIAM J, Comput., 29 (1999), pp. 336-350] and by Alon and Sudakov [Combin. Probab. Comput., 9 (2000), pp. 1-12]. Here we further extend their results and show, for the first time, that the local analyses of the Goemans and Williamson MAX CUT algorithm, as well as its extension by Zwick, are tight for every possible relative size of the maximum cut in the sense that the expected value of the solutions obtained by the algorithms may be as small as the analyses ensure. We also obtain similar results for a related problem. Our approach is quite general and could possibly be applied to some additional problems and algorithms.
A new efficient two-dimensional warping algorithm is presented, in which sub-optimal warping is attained by iterating DP-based local optimization of warp on partially overlapping subplane sequence. From an experimenta...
详细信息
A new efficient two-dimensional warping algorithm is presented, in which sub-optimal warping is attained by iterating DP-based local optimization of warp on partially overlapping subplane sequence. From an experimental comparison with a conventional approximation algorithm based on beam search DP, relative superiority of the proposed algorithm is established.
Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!/p(n) tours for some polynomial p(n) for every...
详细信息
Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!/p(n) tours for some polynomial p(n) for every TSP instance on n cities. They conjectured that, unless P = NP, the answer to this question is negative. We prove that the answer to this question is, in fact, positive. A generalization of the TSP, the quadratic assignment problem, is also considered with respect to the analogous question. Probabilistic, graph-theoretical, group-theoretical and number-theoretical methods and results are used. (C) 2002 Elsevier Science B.V. All rights reserved.
We extend the classic notion of well-separated pair decomposition [10] to the (weighted) unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the ...
详细信息
ISBN:
(纸本)9781581136746
We extend the classic notion of well-separated pair decomposition [10] to the (weighted) unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c≥1, there exists a c-well-separated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where k≥3, there exists a c-well-separated pair decomposition with O(n2-2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric.
Gu et al. gave a 2-approximation for computing the minimal number of inversions and transpositions needed to sort a permutation. There is evidence that, from the point of view of computational molecular biology, a mor...
详细信息
Gu et al. gave a 2-approximation for computing the minimal number of inversions and transpositions needed to sort a permutation. There is evidence that, from the point of view of computational molecular biology, a more adequate objective function is obtained, if transpositions are given double weight. We present a (1 + epsilon)-approximation for this problem, based on the exact algorithm of Hannenhalli and Pevzner for sorting by reversals only. (C) 2002 Elsevier Science B.V. All rights reserved.
We study the problem of finding an acyclic orientation of an undirected graph, such that each (oriented) path is covered by a limited number k of maximal cliques. This is equivalent to finding a k-approximate solution...
详细信息
We study the problem of finding an acyclic orientation of an undirected graph, such that each (oriented) path is covered by a limited number k of maximal cliques. This is equivalent to finding a k-approximate solution for the interval coloring problem on a graph. We focus our attention on claw-free chordal graphs, and show how to find an orientation of such a graph in linear time, which guarantees that each path is covered by at most two maximal cliques. This extends previous published results on other graph classes where stronger assumptions were made. (C) 2002 Elsevier Science B.V. All rights reserved.
暂无评论