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.
We consider the a priori traveling repairman problem, which is a stochastic version of the classic traveling repairman problem. Given a metric (V, d) with a root r is an element of V, the traveling repairman problem (...
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We consider the a priori traveling repairman problem, which is a stochastic version of the classic traveling repairman problem. Given a metric (V, d) with a root r is an element of V, the traveling repairman problem (TRP) involves finding a tour originating from r that minimizes the sum of arrival-times at all vertices. In its a priori version, we are also given independent probabilities of each vertex being active. We want to find a master tour tau originating from r and visiting all vertices. The objective is to minimize the expected sum of arrival-times at all active vertices, when tau is shortcut over the inactive vertices. We obtain the first constant-factor approximation algorithm for a priori TRP under independent non-uniform probabilities. Our result provides a general reduction from non-uniform to uniform probabilities, and uses (in black-box manner) an existing approximation algorithm for a priori TRP under uniform probabilities. (C) 2020 Elsevier B.V. All rights reserved.
We develop the first approximation algorithm with worst-case performance guarantee for capacitated stochastic periodic-review inventory systems with setup costs. The structure of the optimal control policy for such sy...
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We develop the first approximation algorithm with worst-case performance guarantee for capacitated stochastic periodic-review inventory systems with setup costs. The structure of the optimal control policy for such systems is extremely complicated, and indeed, only some partial characterization is available. Thus, finding provably near-optimal control policies has been an open challenge. In this article, we construct computationally efficient approximate optimal policies for these systems whose demands can be nonstationary and/or correlated over time, and show that these policies have a worst-case performance guarantee of 4. We demonstrate through extensive numerical studies that the policies empirically perform well, and they are significantly better than the theoretical worst-case guarantees. We also extend the analyses and results to the case with batch ordering constraints, where the order size has to be an integer multiple of a base load. (C) 2014 Wiley Periodicals, Inc.
Cellular networks are generally modeled as node-weighted graphs, where the nodes represent cells and the edges represent the possibility of radio interference. An algorithm for the channel assignment problem must assi...
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Cellular networks are generally modeled as node-weighted graphs, where the nodes represent cells and the edges represent the possibility of radio interference. An algorithm for the channel assignment problem must assign as many channels as the weight indicates to every node, such that any two channels assigned to the same node satisfy the co-site constraint, and any two channels assigned to adjacent nodes satisfy the inter-site constraint. We describe several approximation algorithms for channel assignment with arbitrary co-site and inter-site constraints for odd cycles and the so-called hexagon graphs that are often used to model cellular networks. The algorithms given for odd cycles are optimal for some values of constraints, and have performance ratio at most 1 + 1/(n - 1) for all other cases, where n is the length of the cycle. Our main result is an algorithm of performance ratio at most 4/3 + 1/100 for hexagon graphs with arbitrary co-site and inter-site constraints. (C) 2001 Elsevier Science B.V. All rights reserved.
In this paper, we consider the classical problem of link scheduling in wireless networks under an accurate interference model, in which correct packet reception at a receiver node depends on the signa-to-interference-...
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In this paper, we consider the classical problem of link scheduling in wireless networks under an accurate interference model, in which correct packet reception at a receiver node depends on the signa-to-interference-plus-noise ratio (SINR). While most previous work on wireless networks has addressed the scheduling problem using simplistic graph-based or distance-based interference models, a few recent papers have investigated scheduling with SINR-based interference models. However, these papers have either used approximations to the SINR model or have ignored important aspects of the problem. We study the problem of wireless link scheduling under the exact SINR model and present the first known true approximation algorithms for transmission scheduling under the exact model. We also introduce an algorithm with a proven approximation bound with respect to the length of the optimal schedule under primary interference. As an aside, our study identifies a class of "difficult to schedule" links, which hinder the derivation of tighter approximation bounds. Furthermore, we characterize conditions under which scheduling under SINR-based interference is within a constant factor from optimal under primary interference, which implies that secondary interference only degrades performance by a constant factor in these situations.
This paper describes a general technique that can be used to obtain approximation schemes for various NP-complete problems on planar graphs. The strategy depends on decomposing a planar graph into subgraphs of a form ...
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This paper describes a general technique that can be used to obtain approximation schemes for various NP-complete problems on planar graphs. The strategy depends on decomposing a planar graph into subgraphs of a form we call k-outerplanar. For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum independent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least k/(k + 1) optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k + 1)/k optimal. Taking k = inverted right perpendicular c log log n inverted left perpendicular or k = right perpendicular c log n left perpendicular, where n is the number of nodes and c is some constant, we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar graphs for which the problems are known to be solvable in polynomial time.
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
In this article we focus on approximation algorithms for facility location problems with subadditive costs. As examples of such problems, we present three facility location problems with stochastic demand and exponent...
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In this article we focus on approximation algorithms for facility location problems with subadditive costs. As examples of such problems, we present three facility location problems with stochastic demand and exponential servers, respectively inventory. We present a (1 + epsilon, 1)-reduction of the facility location problem with subadditive costs to the soft capacitated facility location problem, which implies the existence of a 2(1 + epsilon) -approximation algorithm. For a special subclass of subadditive functions, we obtain a 2-approximation algorithm by reduction to the linear cost facility location problem. (C) 2006 Elsevier B.V. All rights reserved.
Sorting permutations with various operations has applications in macro rearrangement of genes in a genome and the design of computer interconnection networks. Block-interchange is a powerful operation that swaps two s...
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Sorting permutations with various operations has applications in macro rearrangement of genes in a genome and the design of computer interconnection networks. Block-interchange is a powerful operation that swaps two substrings that are called as blocks in literature, in a given permutation. When the blocks are restricted to be adjacent then one obtains a well studied operation: transposition. We call either a prefix or a suffix as an extreme. Restricting one of the swapped blocks to be an extreme in block-interchange operation yields a prefix or a suffix block-interchange respectively, the two types of extreme block-interchanges. For prefix block-interchange operation over permutations we design: (i) an optimum algorithm to sort reverse permutation, R-n, in n/2 moves, (ii) a simple 2-approximation algorithm, and (iii) for permutations with O(1) cycles, a 4/3 approximation algorithm. Due to symmetry, these results apply to suffix block-interchange operation also. (C) 2021 Elsevier B.V. All rights reserved.
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