In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An active line of resea...
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In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified;this issue is one of the principal motivations for the framework of Markov random fields, and it arises in areas such as image processing, biometry, and document analysis. In its most basic form, this style of analysis seeks to find a classification that optimizes a combinatorial function consisting of assignment costs-based on the individual choice of label we make for each object-and separation costs-based on the pair of choices we make for two "related" objects. We formulate a general classification problem of this type, the metric labeling problem;we show that it contains as special cases a number of standard classification frameworks, including several arising from the theory of Markov random fields. From the perspective of combinatorial optimization, our problem can be viewed as a substantial generalization of the multiway cut problem, and equivalent to a type of uncapacitated quadratic assignment problem. We provide the first nontrivial polynomial-time approximation algorithms for a general family of classification problems of this type. Our main result is an O (log k log log k)-approximation algorithm for the metric labeling problem, with respect to an arbitrary metric on a set of k labels, and an arbitrary weighted graph of relationships on a set of objects. For the special case in which the labels are endowed with the uniform metric-all distances are the same-our methods provide a 2-approximation algorithm.
In this article we study a generalized team orienteering problem (GTOP), which is to find service paths for multiple homogeneous vehicles in a network such that the profit sum of serving the nodes in the paths is maxi...
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In this article we study a generalized team orienteering problem (GTOP), which is to find service paths for multiple homogeneous vehicles in a network such that the profit sum of serving the nodes in the paths is maximized, subject to the cost budget of each vehicle. This problem has many potential applications in IoTs and smart cities, such as dispatching energy-constrained mobile chargers to charge as many energy-critical sensors as possible to prolong the network lifetime. In this article, we first formulate the GTOP problem, where each node can be served by different vehicles, and the profit of serving the node is a submodular function of the number of vehicles serving it. We then propose a novel (1-(1/epsilon)1/2+epsilon)-approximation algorithm for the problem, where epsilon is a given constant with 0 < epsilon <= 1 and e is the base of the natural logarithm. In particular, the approximation ratio is about 0.33 when epsilon = 0.5. In addition, we devise an improved approximation algorithm for a special case of the problem where the profit is the same by serving a node once and multiple times. We finally evaluate the proposed algorithms with simulation experiments, and the results of which are very promising. Especially, the profit sums delivered by the proposed algorithms are up to 14% higher than those by existing algorithms, and about 93.6% of the optimal solutions.
Given a single machine and a set of jobs with due dates, the classical NP-hard. problem of scheduling to minimize total tardiness is a well-understood one. Lawler gave a fully polynomial-time approximation scheme (FPT...
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Given a single machine and a set of jobs with due dates, the classical NP-hard. problem of scheduling to minimize total tardiness is a well-understood one. Lawler gave a fully polynomial-time approximation scheme (FPTAS) for it some 20 years ago. If the jobs have positive weights the problem of minimizing total weighted tardiness seems to be considerably more intricate. it. In this paper, we give some of the first approximation algorithms for it. We examine first the weighted problem with a fixed number of due dates and we design a pseudopolynomial algorithm for it. We show how to transform the pseudopolynomial algorithm to an FPTAS for the case where the weights are polynomially bounded. For the case with an arbitrary number of due dates and polynomially bounded processing times, we provide a quasipolynomial algorithm which produces a schedule whose value has an additive error proportional to the weighted sum of the due dates. We also investigate the performance of algorithms for minimizing the related total weighted late work objective. (c) 2006 Elsevier B.V. All rights reserved.
Broadcasting is a fundamental operation in wireless networks and plays an important role in the communication protocol design. In multihop wireless networks, however, interference at a node due to simultaneous transmi...
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Broadcasting is a fundamental operation in wireless networks and plays an important role in the communication protocol design. In multihop wireless networks, however, interference at a node due to simultaneous transmissions from its neighbors makes it nontrivial to design a minimum-latency broadcast algorithm, which is known to be NP-complete. We present a simple 12-approximation algorithm for the one-to-all broadcast problem that improves all previously known guarantees for this problem. We then consider the all-to-all broadcast problem where each node sends its own message to all other nodes. For the all-to-all broadcast problem, we present two algorithms with approximation ratios of 20 and 34, improving the best result available in the literature. Finally, we report experimental evaluation of our algorithms. Our studies indicate that our algorithms perform much better in practice than the worst-case guarantees provided in the theoretical analysis and achieve up to 37 percent performance improvement over existing schemes.
We study a capacitated network design problem with applications in local access network design. Given a network, the problem is to route flow from several sources to a sink and to install capacity on the edges to supp...
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We study a capacitated network design problem with applications in local access network design. Given a network, the problem is to route flow from several sources to a sink and to install capacity on the edges to support the flow at minimum cost. Capacity can be purchased only in multiples of a fixed quantity. All the flow from a source must be routed in a single path to the sink. This NP-hard problem generalizes the Steiner tree problem and also more effectively models the applications traditionally formulated as capacitated tree problems. We present an approximation algorithm with performance ratio (rho(ST) + 2) where rho(ST) is the performance ratio of any approximation algorithm for the minimum Steiner tree problem. When all sources have unit demand, the ratio improves to (rho(ST) + 1) and, in particular, to 2 when all nodes in the graph are sources.
A number of recent papers on approximation algorithms have used the square roots of unity, -1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit ve...
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A number of recent papers on approximation algorithms have used the square roots of unity, -1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit vectors in real space using semidefinite programming in order to obtain near optimum solutions to these problems. In this paper, we consider using the cube roots of unity, 1, e(i2pi/3), and e(i4pi/3), to represent ternary decision variables for problems in combinatorial optimization. Here the natural relaxation is that of unit vectors in complex space. We use an extension of semidefinite programming to complex space to solve the natural relaxation, and use a natural extension of the random hyperplane technique introduced by the authors in Goemans and Williamson (J. ACM 42 (1995) 1115-1145) to obtain near-optimum solutions to the problems. In particular, we consider the problem of maximizing the total weight of satisfied equations x(u) - x(v) drop c (mod 3) and inequations x - x(v) not equivalent to c (mod 3), where x(u) epsilon {0, 1, 2} for all u. This problem can be used to model the MAx-3-CUT problem and a directed variant we call MAX-3-DICUT. For the general problem, we obtain a 0.793733-approximation algorithm. If the instance contains only inequations (as it does for MAX-3-CUT), we obtain a performance guarantee of (7)/(12) + (3)/(4pi2) arccos(2)(- 1/4) - epsilon>0.836008. This compares with proven performance guarantees of 0.800217 for MAX-3-CUT (by Frieze and Jerrum (Algorithmica 18 (1997) 67-81) and 1 + 10(-8) for the general problem (by Andersson et al. (J. algorithms 3 39 (2001) 162-204)). It matches the guarantee of 0.836008 for MAX-3-CUT found independently by de Klerk et al. (On approximate graph colouring and Max-k-Cut algorithms based on the 9-function, Manuscript, October 2000). We show that all these algorithms are in fact equivalent in the case of MAX-3CUT, and that our algorithm is the same as that of Andersson et al. in the more gener
Stochastic combinatorial optimization problems are usually defined as planning problems, which involve purchasing and allocating resources in order to meet uncertain needs. For example, network designers need to make ...
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Stochastic combinatorial optimization problems are usually defined as planning problems, which involve purchasing and allocating resources in order to meet uncertain needs. For example, network designers need to make their best guess about the future needs of the network and purchase capabilities accordingly. Facing uncertain in the future, we either "wait and see" changes, or postpone decisions about resource allocation until the requirements or constraints become realized. Specifically, in the field of stochastic combinatorial optimization, some inputs of the problems are uncertain, but follow known probability distributions. Our goal is to find a strategy that minimizes the expected cost. In this paper, we consider the two-stage finite-scenario stochastic set cover problem and the single sink rent-or-buy problem by presenting primal-dual based approximation algorithms for these two problems with approximation ratio 2 eta and 4.39, respectively, where eta is the maximum frequency of the element of the ground set in the set cover problem.
Intensity modulated radiation therapy (IMRT) is one of the most effective modalities for modem cancer treatment. The key to successful IMRT treatment hinges on the delivery of a two-dimensional discrete radiation inte...
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Intensity modulated radiation therapy (IMRT) is one of the most effective modalities for modem cancer treatment. The key to successful IMRT treatment hinges on the delivery of a two-dimensional discrete radiation intensity matrix using a device called a multileaf collimator (MLC). Mathematically, the delivery of an intensity matrix using an MLC can be viewed as the problem of representing a non-negative integral matrix (i.e., the intensity matrix) by a linear combination of certain special non-negative integral matrices called segments, where each such segment corresponds to one of the allowed states of the MLC. The problem of representing the intensity matrix with the minimum number of segments is known to be NP-complete. In this paper, we present two approximation algorithms for this matrix representation problem. To the best of our knowledge, these are the first algorithms to achieve non-trivial performance guarantees for multi-row intensity matrices. (c) 2006 Elsevier B.V. All rights reserved.
In the Workload Partition Problem (WPP) we are given a set of n jobs to be scheduled on a set of m identical parallel machines. Each job has its own workload and the scheduling cost on each machine is a convex functio...
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In the Workload Partition Problem (WPP) we are given a set of n jobs to be scheduled on a set of m identical parallel machines. Each job has its own workload and the scheduling cost on each machine is a convex function of the total worldoad of the jobs assigned to it. The objective is to minimize the total cost on the set of m machines. Shabtay and Kaspi (2006) showed that the WPP is equivalent to a scheduling problem on m identical machines with controllable processing times and with the scheduling criterion of minimizing the makespan. They also proved that the WPP is NP-hard when in = 2. However, they left as an open question whether the problem is ordinary or strongly NP-hard. Moreover, they provided no practical tools to solve the problem. We bridge those gaps in the literature by showing that the WWP problem is strongly NP-hard when m is part of the input. Furthermore, we present two different approximation algorithms for solving the MAT problem. The first one is a fully polynomial time approximation scheme (FPTAS) for a fixed number of machines, while the second is a modification of the well-known longest processing time (LPT) heuristic. We show that our modified LPT heuristic guarantees a solution with a constant approximation ratio, whose value depends on the instance parameters. Crown Copyright (C) 2016 Published by Elsevier B.V. All rights reserved.
We study the problem of throughput maximization in multihop wireless networks with end-to-end delay constraints for each session. This problem has received much attention starting with the work of Grossglauser and Tse...
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We study the problem of throughput maximization in multihop wireless networks with end-to-end delay constraints for each session. This problem has received much attention starting with the work of Grossglauser and Tse (2002), and it has been shown that there is a significant tradeoff between the end-to-end delays and the total achievable rate. We develop algorithms to compute such tradeoffs with provable performance guarantees for arbitrary instances, with general interference models. Given a target delay-bound Delta(c) for each session c, our algorithm gives a stable flow vector with a total throughput within a factor of O(log Delta(m)/ log log Delta(m)) of the maximum, so that the per-session (end-to-end) delay is, O(((log Delta(m)/ log log Delta(m))Delta(c))(2)) where Delta(m) = max(c){Delta(c)};note that these bounds depend only on the delays, and not on the network size, and this is the first such result, to our knowledge.
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