In the cutting stock problem, we are given a set of objects of different types, and the goal is to pack them all in the minimum possible number of identical bins. All objects have integer lengths, and objects of diffe...
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In the cutting stock problem, we are given a set of objects of different types, and the goal is to pack them all in the minimum possible number of identical bins. All objects have integer lengths, and objects of different types have different sizes. The total length of the objects packed in a bin cannot exceed the capacity of the bin. In this paper, we consider the version of the problem in which the number of different object types is constant, and we present a polynomial-time algorithm that computes a solution using at most one more bin than an optimum solution.
We develop a new dependent randomized rounding method for approximation of a number of optimization problems with integral assignment constraints. The core of the method is a simple, intuitive, and computationally eff...
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We develop a new dependent randomized rounding method for approximation of a number of optimization problems with integral assignment constraints. The core of the method is a simple, intuitive, and computationally efficient geometric rounding that simultaneously rounds multiple points in a multi-dimensional simplex to its vertices. Using this method we obtain in a systematic way known as well as new results for the hub location, metric labeling, winner determination and consistent labeling problems. A comprehensive comparison to the dependent randomized rounding method developed by Kleinberg and Tardos (J. ACM 49(5):616-639, 2002) and its variants is also conducted. Overall, our geometric rounding provides a simple and effective alternative for rounding various integer optimization problems.
The complexity of the Bandpass problem is re-investigated. Specifically, we show that the problem with any fixed bandpass number Ba parts per thousand yen2 is NP-hard. Next, a row stacking algorithm is proposed for th...
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The complexity of the Bandpass problem is re-investigated. Specifically, we show that the problem with any fixed bandpass number Ba parts per thousand yen2 is NP-hard. Next, a row stacking algorithm is proposed for the problem with three columns, which produces a solution that is at most 1 less than the optimum. For the special case B=2, the row stacking algorithm guarantees an optimal solution. On approximation, for the general problem, we present an O(B (2))-algorithm, which reduces to a 2-approximation algorithm for the special case B=2.
In this paper, we study the INTERFERENCE-AWARE BROADCAST SCHEDULING problem, where all nodes in the Euclidean plane have a transmission range and an interference range equal to r and alpha r for alpha >= 1, respect...
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In this paper, we study the INTERFERENCE-AWARE BROADCAST SCHEDULING problem, where all nodes in the Euclidean plane have a transmission range and an interference range equal to r and alpha r for alpha >= 1, respectively. Minimizing latency is known to be NP-Hard even when alpha = 1. The network radius D, the maximum graph distance from the source to any node, is also known to be a lower bound. We formulate the problem as integer programs (IP) and optimally solve moderate-size instances. We also propose six variations of heuristics, which require no pre-processing of inputs, based on the number of receivers gained by each additional simultaneous transmitting node. The experimental results show that the best heuristics give solutions that exceed the optimal solutions by only 13-20%. Further, an O(alpha D) schedule is proven to exist yielding an O(alpha) approximation algorithm. (C) 2010 Elsevier B.V. All rights reserved.
We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph G, where a nonnegative weight is assigned to each edge of G and ...
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We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph G, where a nonnegative weight is assigned to each edge of G and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Several heuristics have been proposed to tackle this NP-hard problem, which has some interesting applications. In this paper we show that this problem is APX-hard, even when restricted to unweighted graphs, and hence does not admit a polynomial-time approximation scheme, unless P = NP. Using a recent result on the approximability of lower-stretch spanning trees (Elkin et al. (2005) [7]), we obtain that the problem is approximable within O(log(2) n log log n) for arbitrary graphs. We obtain tighter approximability bounds for dense graphs. In particular, the problem restricted to complete graphs admits a polynomial-time approximation scheme. (C) 2010 Elsevier B.V. All rights reserved.
We address the problem of partitioning a set of independent, periodic, real-time tasks over a fixed set of heterogeneous processors while minimizing the energy consumption of the computing platform subject to a guaran...
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We address the problem of partitioning a set of independent, periodic, real-time tasks over a fixed set of heterogeneous processors while minimizing the energy consumption of the computing platform subject to a guaranteed quality of service requirement. This problem is NP-hard and we present a fully polynomial time approximation scheme for this problem. The main contribution of our work is in tackling the problem in a completely discrete, and possibly arbitrarily structured, setting. In other words, each processor has a discrete set of speed choices. Each task has a computation time that is dependent on the processor that is chosen to execute the task and on the speed at which that processor is operated. Further, the energy consumption of the system is dependent on the decisions regarding task allocation and speed settings. (c) 2011 Elsevier Inc. All rights reserved.
This paper considers a two-stage production scheduling problem in which each activity requires two operations to be processed in stages 1 and 2, respectively. There are two options for processing each operation: the f...
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This paper considers a two-stage production scheduling problem in which each activity requires two operations to be processed in stages 1 and 2, respectively. There are two options for processing each operation: the first is to produce it by utilizing in-house resources, while the second is to outsource it to a subcontractor. For in-house operations, a schedule is constructed and its performance is measured by the makespan, that is, the latest completion time of operations processed in-house. Operations by subcontractors are instantaneous but require outsourcing cost. The objective is to minimize the weighted sum of the makespan and the total outsourcing cost. This paper analyzes how the model's computational complexity changes according to unit outsourcing costs in both stages and describes the boundary between NP-hard and polynomially solvable cases. Finally, this paper presents an approximation algorithm for one NP-hard case. (C) 2011 Elsevier B.V. All rights reserved.
We consider a generalization of the classical open shop and flow shop scheduling problems where the jobs are located at the vertices of an undirected graph and the machines, initially located at the same vertex, have ...
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We consider a generalization of the classical open shop and flow shop scheduling problems where the jobs are located at the vertices of an undirected graph and the machines, initially located at the same vertex, have to travel along the graph to process the jobs. The objective is to minimize the makespan. In the tour-version the makespan means the time by which each machine has processed all jobs and returned to the initial location. While in the path-version the makespan represents the maximum completion time of the jobs. We present improved approximation algorithms for various cases of the open shop problem on a general graph, and the tour-version of the two-machine flow shop problem on a tree. Also, we prove that both versions of the latter problem are NP-hard, which answers an open question posed in the literature. (C) 2011 Elsevier B.V. All rights reserved.
In wireless sensor networks, when each target is covered by multiple sensors, sensors can take turns to monitor the targets in order to extend the lifetime of the network. In this paper, we address how to improve netw...
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In wireless sensor networks, when each target is covered by multiple sensors, sensors can take turns to monitor the targets in order to extend the lifetime of the network. In this paper, we address how to improve network lifetime through optimal scheduling of sensor nodes. We present two algorithms to achieve the maximum lifetime while maintaining the required coverage: a linear programming-based exponential-time exact solution, and an approximation algorithm. Numerical simulation results from the approximation algorithm are compared to the exact solution and show a high degree of accuracy and efficiency.
Let G = (V, E) be a connected graph such that each edge e is an element of E is weighted by a nonnegative real w(e). Let s be a vertex designated as a sink, M subset of V be a set of terminals with a demand function q...
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Let G = (V, E) be a connected graph such that each edge e is an element of E is weighted by a nonnegative real w(e). Let s be a vertex designated as a sink, M subset of V be a set of terminals with a demand function q : M -> R+, kappa > 0 be a routing capacity, and lambda >= 1 be an integer edge capacity. The capacitated tree-routing problem (CTR) asks to find a partition M = {Z(1), Z(2), ... , Z(l)} of M and a set tau = {T-1, T-2, ... , T-l} of trees of G such that each T-i contains Z(i) boolean OR {s} and satisfies Sigma(upsilon is an element of Zi) q(upsilon) <= kappa. A single copy of an edge e is an element of E can be shared by at most lambda trees in tau;any integer number of copies of e are allowed to be installed, where the cost of installing a copy of e is w(e). The objective is to find a solution (M, T) that minimizes the total installing cost. In this paper, we propose a (2+rho(ST))-approximation algorithm to CTR, where rho(ST) is any approximation ratio achievable for the Steiner tree problem.
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