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 consider the problem of constructing optimal decision trees: given a collection of tests that can disambiguate between a set of m possible diseases, each test having a cost, and the a priori likelihood of any parti...
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We consider the problem of constructing optimal decision trees: given a collection of tests that can disambiguate between a set of m possible diseases, each test having a cost, and the a priori likelihood of any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? This problem has been studied in several works, with O(log m)-approximations known in the special cases when either costs or probabilities are uniform. In this paper, we settle the approximability of the general problem by giving a tight O(log m)-approximation algorithm. We also consider a substantial generalization, the adaptive traveling salesman problem. Given an underlying metric space, a random subset S of vertices is drawn from a known distribution, but S is initially unknown-we get information about whether any vertex is in S only when it is visited. What is a good adaptive strategy to visit all vertices in the random subset S while minimizing the expected distance traveled? This problem has applications in routing message ferries in ad hoc networks and also models switching costs between tests in the optimal decision tree problem. We give a polylogarithmic approximation algorithm for adaptive TSP, which is nearly best possible due to a connection to the well-known group Steiner tree problem. Finally, we consider the related adaptive traveling repairman problem, where the goal is to compute an adaptive tour minimizing the expected sum of arrival times of vertices in the random subset S;we obtain a polylogarithmic approximation algorithm for this problem as well.
In generalized tree alignment problem, we are given a set S of k biologically related sequences and we are interested in a minimum cost evolutionary tree for S. In many instances of this problem partial phylogenetic t...
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In generalized tree alignment problem, we are given a set S of k biologically related sequences and we are interested in a minimum cost evolutionary tree for S. In many instances of this problem partial phylogenetic tree for S is known. In such instances, we would like to make use of this knowledge to restrict the tree topologies that we consider and construct a biologically relevant minimum cost evolutionary tree. So, we propose the following natural generalization of the generalized tree alignment problem, a problem known to be MAX-SNP Hard, stated as follows: Constrained Generalized Tree Alignment Problem [S. Divakaran, algorithms and heuristics for constrained generalized alignment problem, DIMACS Technical Report 2007-21, 2007]: Given a set S of k related sequences and a phylogenetic forest comprising of node-disjoint phylogenetic trees that specify the topological constraints that an evolutionary tree of S needs to satisfy, construct a minimum cost evolutionary tree for S. In this paper, we present constant approximation algorithms for the constrained generalized tree alignment problem. For the generalized tree alignment problem, a special case of this problem, our algorithms provide a guaranteed error bound of 2 - 2/k. (C) 2009 Published by Elsevier B.V.
The input to the METRIC MAXIMUM CLUSTERING PROBLEM WITH GIVEN CLUSTER SIZES consists of a complete graph G=(V, E) with edge weights satisfying the triangle inequality, and integers c(1),...., c(p) that sum to I V. The...
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The input to the METRIC MAXIMUM CLUSTERING PROBLEM WITH GIVEN CLUSTER SIZES consists of a complete graph G=(V, E) with edge weights satisfying the triangle inequality, and integers c(1),...., c(p) that sum to I V. The goal is to find a partition of V into disjoint clusters of sizes c(1),....,c(p), that maximizes the sum of weights of edges whose two ends belong to the same cluster. We describe approximation algorithms for this problem. (C) 2003 Elsevier Science 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.
We consider the problem of scheduling a set of jobs on a single machine subject to inventory constraints, i.e., conditions that jobs add or remove items to or from a centralized inventory, respectively. Jobs that remo...
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We consider the problem of scheduling a set of jobs on a single machine subject to inventory constraints, i.e., conditions that jobs add or remove items to or from a centralized inventory, respectively. Jobs that remove items cannot be processed if the required number of items is not available. We focus on scheduling problems on a single machine where the objective is to minimize the total weighted completion time. In this paper, we design 2-approximation algorithms for special cases of the problem that run in polynomial time.
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.
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