The paper considers the job shop scheduling problem to minimize the makespan. It is assumed that each job consists of at most two operations, one of which is to be processed on one of m greater than or equal to 2 mach...
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The paper considers the job shop scheduling problem to minimize the makespan. It is assumed that each job consists of at most two operations, one of which is to be processed on one of m greater than or equal to 2 machines, while the other operation must be performed on a single bottleneck machine, the same for all jobs. For this strongly NP-hard problem we present two heuristics with improved worst-case performance. One of them guarantees a worst-case performance ratio of 3/2. The other algorithm creates a schedule with the makespan that exceeds the largest machine workload by at most the length of the largest operation. (C) 2000 Elsevier Science B.V. All rights reserved.
We consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion limes. This problem is NP-hard in the strong sense. The best approximation re...
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We consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion limes. This problem is NP-hard in the strong sense. The best approximation result known so far was a 1/2 (1 + root 2)-approximation algorithm that has been derived by Kawaguchi and Kyan back in 1986. The contribution of this paper is a polynomial time approximation scheme for this setting, which settles a problem that was open for a long time. Moreover, our result constitutes the first known approximation scheme for a strongly NP-hard scheduling problem with minsum objective.
This payer considers generalized 2SAT problems, MAX GEN2SAT and MIN GEN2SAT. Instances of these problems are defined on a collection of "clauses", which we refer to as genclauses. A genclause is any boolean ...
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This payer considers generalized 2SAT problems, MAX GEN2SAT and MIN GEN2SAT. Instances of these problems are defined on a collection of "clauses", which we refer to as genclauses. A genclause is any boolean function on two variables, and each genclause has a non-negative weight associated with it in the problems that are considered. The objective of MAX GEN2SAT (MIN GEN2SAT) is to select a truth assignment that maximizes (minimizes) the total weight of satisfied genclauses. Goemans and Williamson (J. ACM 42(6) (1995) 1115-1145) used semidefinite programming and were able to provide substantial improvements in approximation factor guarantee for several important problems: MAX 2SAT, MAX CUT, MAX DICUT. In this paper we show how their approximation technique can be used to yield an approximation algorithm for MAX GEN2SAT, for which MAX 2SAT, MAX CUT, MAX DICUT are special cases. For MIN GEN2SAT, employing a recent technique of Hochbaum (Manuscript, UC Berkeley, June 1997) leads to easy recognition of which instances are polynomial or 2-approximable. The polynomial instances of MIN GEN2SAT have corresponding instances of MAX GEN2SAT which are thus identified as solvable in polynomial time. Among the applications of the approximation algorithms described it is shown that the forest harvesting problem has a 0.87856-approximation algorithm. (C) 2000 Elsevier Science B.V. All rights reserved.
We consider the problem of scheduling a set of classes to classrooms with the objective of minimizing the number of classrooms used. The major constraint that we must obey is that no two classes can be assigned to the...
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We consider the problem of scheduling a set of classes to classrooms with the objective of minimizing the number of classrooms used. The major constraint that we must obey is that no two classes can be assigned to the same classroom at the same time on the same day of the week. We present an algorithm that produces a nearly optimal schedule for an arbitrary set of classes. The algorithm's first stage produces a packing of classes using a combination of a greedy algorithm and a non-bipartite matching and the second stage consists of a bipartite matching. First we show that for one variant of the problem our algorithm produces schedules that require a number of classrooms that is always within a small additive constant of optimal. Then we show that for an interesting variant of the problem the same algorithm produces schedules that require a small constant factor more classrooms than optimal. Finally, we report on experimental results of our algorithm using actual data and also show how to create schedules with other desirable characteristics.
In this work we consider the maximum p-facility location problem with k additional resource constraints. We prove that, the simple greedy algorithm has performance guarantee (1 - e(-(k+1)))/(k + 1). In the case k = 0 ...
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In this work we consider the maximum p-facility location problem with k additional resource constraints. We prove that, the simple greedy algorithm has performance guarantee (1 - e(-(k+1)))/(k + 1). In the case k = 0 our performance guarantee coincides with bound due to [4]. (C) 2000 Elsevier Science B.V. All rights reserved.
Consider the following problem: Given an undirected graph with nonnegative edge costs and requirements k(u) for every node u, find a minimum-cost subgraph that contains max{k(u), k(v)} internally disjoint paths betwee...
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Consider the following problem: Given an undirected graph with nonnegative edge costs and requirements k(u) for every node u, find a minimum-cost subgraph that contains max{k(u), k(v)} internally disjoint paths between every pair of nodes u, v. For k = max k(u) greater than or equal to 2, this problem is NP-hard. The best-known algorithm for it has an approximation ratio of 2(k - 1). For a general instance of the problem, for no value of k greater than or equal to 2, a better approximation algorithm was known. We consider the case of small requirements k(u) epsilon {1, 2, 3};these may arise in applications, as, in practical networks, the connectivity requirements are usually rather small. For this case, we give an algorithm with an approximation ratio of 10/3. This improves the best previously known approximation ratio of 4. Our algorithm also implies an improvement for arbitrary k. In the case in which we have an initial graph which is 2-connected, our algorithm achieves an approximation ratio of 2. (C) 2000 John Wiley & Sons, Inc.
We present a linear-time algorithm that decomposes a convex polygon conformally into a minimum number of strictly convex quadrilaterals. Moreover, we characterise the polygons that can be decomposed without additional...
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We present a linear-time algorithm that decomposes a convex polygon conformally into a minimum number of strictly convex quadrilaterals. Moreover, we characterise the polygons that can be decomposed without additional vertices inside the polygon, and we present a linear-time algorithm for such decompositions, too. As an application, we consider the problem of constructing a minimum conformal refinement of a mesh in the three-dimensional space, which approximates the surface of a workpiece. We prove that this problem is strongly Np-hard, and we present a linear-time algorithm with a constant approximation ratio of four.
We consider a problem that arises from communication in all-optical networks. Data are transmitted from source nodes to destination nodes via fixed routes. The high bandwidth of the optic fiber allows for wavelength-d...
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We consider a problem that arises from communication in all-optical networks. Data are transmitted from source nodes to destination nodes via fixed routes. The high bandwidth of the optic fiber allows for wavelength-division multiplexing so that a single physical optical link can carry several logical signals of different wavelengths. The problem is to carry out a set of requests using a limited number of wavelengths so that different routes using the same wavelength never use the same physical link. We focus on trees of rings which are constructed as follows: Start from a tree and replace each node of the tree by a cycle. Each edge in the tree corresponds to the corresponding cycles sharing a common node. We design an approximation algorithm that routes any set of requests on the tree of rings using no more than 2.5w(opt) wavelengths, where w(opt) is the minimum possible number of wavelengths for that set of requests. This improves a 3-approximation solution of Raghavan and Upfal. (C) 2000 John Wiley & Sons, Inc.
We present a simple and unified approach for developing and analyzing approximation algorithms for covering problems. We illustrate this on approximation algorithms for the following problems: Vertex Cover, Set Cover,...
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We present a simple and unified approach for developing and analyzing approximation algorithms for covering problems. We illustrate this on approximation algorithms for the following problems: Vertex Cover, Set Cover, Feedback Vertex Set, Generalized Steiner Forest, and related problems. The main idea can be phrased as follows: iteratively, pay two dollars (at most) to reduce the total optimum by one dollar (at least), so the rate of payment is no more than twice the rate of the optimum reduction. This implies a total payment (i.e., approximation cost) twice the optimum cost. Our main contribution is based on a formal definition for covering problems, which includes all the above fundamental problems and others. We further extend the, Bafna et al. extension of the Local-Ratio theorem. Our extension eventually yields a short generic r-approximation algorithm which can generate most known approximation algorithms for most covering problems. Another extension of the Local-Ratio theorem to randomized algorithms gives a simple proof of Pitt's randomized approximation for Vertex Cover. Using this approach, we develop a modified greedy algorithm, which for Vertex Cover gives an expected performance ratio less than or equal to 2.
We describe fully polynomial time approximation schemes for various multicommodity ow problems in graphs with m edges and n vertices. We present the rst approximation scheme for maximum multicommodity ow that is indep...
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We describe fully polynomial time approximation schemes for various multicommodity ow problems in graphs with m edges and n vertices. We present the rst approximation scheme for maximum multicommodity ow that is independent of the number of commodities k, and our algorithm improves upon the run time of previous algorithms by this factor of k, running in O* (epsilon (-2)m(2)) time. For maximum concurrent ow and minimum cost concurrent ow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e., k > m/n. Our algorithms build on the framework proposed by Garg and Konemann [ Proceedings of the 39 th Annual IEEE Symposium on Foundations of Computer Science, IEEE, New York, 1998, pp. 300-309]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. The approximation guarantees are obtained by comparison with dual feasible solutions found by our algorithm. Our maximum multicommodity ow algorithm extends to an approximation scheme for the maximum weighted multicommodity ow, which is faster than those implied by previous algorithms by a factor of k/log W, where W is the maximum weight of a commodity.
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