We consider an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provi...
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We consider an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function or that of a reformulated objective function.
Semidefinite relaxations of certain combinatorial optimization problems lead to approximation algorithms with performance guarantees. For large-scale problems, it may not be computationally feasible to solve the semid...
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Semidefinite relaxations of certain combinatorial optimization problems lead to approximation algorithms with performance guarantees. For large-scale problems, it may not be computationally feasible to solve the semidefinite relaxations to optimality. In this paper, we investigate the effect on the performance guarantees of an approximate solution to the semidefinite relaxation for MAxCuT, MAx2SAT, and MAx3SAT. We show that it is possible to make simple modifications to the approximate solutions and obtain performance guarantees that depend linearly on the most negative eigenvalue of the approximate solution, the size of the problem, and the duality gap. In every case, we recover the original performance guarantees in the limit as the solution approaches the optimal solution to the semidefinite relaxation. (C) 2016 Elsevier B.V. All rights reserved.
In the Steiner forest problem, we are given a set of terminal pairs and need to find the minimum cost subgraph that connects each of the terminal pairs together. Motivated by the recent work on greedy approximation al...
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In the Steiner forest problem, we are given a set of terminal pairs and need to find the minimum cost subgraph that connects each of the terminal pairs together. Motivated by the recent work on greedy approximation algorithms for the Steiner forest, we provide efficient implementations of existing approximation algorithms and conduct a thorough experimental study to characterize their performance. We consider several approximation algorithms: the influential primal-dual 2-approximation algorithm due to Agrawal, Klein, and Ravi, the greedy algorithm due to Gupta and Kumar, and a randomized algorithm based on probabilistic approximation by tree metrics. We also consider the simplest heuristic greedy algorithm for the problem, which picks the closest unconnected pair of terminals and connects it using the shortest path between the terminals in the current graph. To characterize the performance of the algorithms, we created a new library with more than one thousand Steiner forest problem instances and conducted an extensive experimental analysis on those instances. Our analysis reveals that for the majority of instances the primal-dual algorithm is the fastest among all the algorithms considered here, and obtains solutions that are very close to the optimal solutions obtained by solving the integer program formulation of the problem.
We consider the single-period joint assortment and inventory planning problem with stochastic demand and dynamic substitution across products, motivated by applications in highly differentiated markets, such as online...
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We consider the single-period joint assortment and inventory planning problem with stochastic demand and dynamic substitution across products, motivated by applications in highly differentiated markets, such as online retailing and airlines. This class of problems is known to be notoriously hard to deal with from a computational standpoint. In fact, prior to the present paper, only a handful of modeling approaches were shown to admit provably good algorithms, at the cost of strong restrictions on customers' choice outcomes. Our main contribution is to provide the first efficient algorithms with provable performance guarantees for a broad class of dynamic assortment optimization models. Under general rank-based choice models, our approximation algorithm is best possible with respect to the price parameters, up to lower-order terms. In particular, we obtain a constant-factor approximation under horizontal differentiation, where product prices are uniform. In more structured settings, where the customers' ranking behavior is motivated by price and quality cues, we derive improved guarantees through tailor-made algorithms. In extensive computational experiments, our approach dominates existing heuristics in terms of revenue performance, as well as in terms of speed, given the myopic nature of our methods. From a technical perspective, we introduce a number of novel algorithmic ideas of independent interest, and unravel hidden relations to submodular maximization.
We present improved approximation algorithms in stochastic optimization. We prove that the multistage stochastic versions of covering integer programs (such as set cover and vertex cover) admit essentially the same ap...
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We present improved approximation algorithms in stochastic optimization. We prove that the multistage stochastic versions of covering integer programs (such as set cover and vertex cover) admit essentially the same approximation algorithms as their standard (nonstochastic) counterparts;this improves upon work of Swamy and Shmoys which shows an approximability that depends multiplicatively on the number of stages. We also present approximation algorithms for facility location and some of its variants in the 2-stage recourse model, improving on previous approximation guarantees. We give a 2.2975-approximation algorithm in the standard polynomial-scenario model and an algorithm with an expected per-scenario 2.4957-approximation guarantee, which is applicable to the more general black-box distribution model.
An approximation algorithm for an optimization problem runs in polynomial time for all instances and is guaranteed to deliver solutions with bounded optimality gap. We derive such algorithms for different variants of ...
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An approximation algorithm for an optimization problem runs in polynomial time for all instances and is guaranteed to deliver solutions with bounded optimality gap. We derive such algorithms for different variants of capacitated location routing, an important generalization of vehicle routing where the cost of opening the depots from which vehicles operate is taken into account. Our results originate from combining algorithms and lower bounds for different relaxations of the original problem; along with location routing we also obtain approximation algorithms for multidepot capacitated vehicle routing by this framework. Moreover, we extend our results to further generalizations of both problems, including a prize-collecting variant, a group version, and a variant where cross-docking is allowed. We finally present a computational study of our approximation algorithm for capacitated location routing on benchmark instances and large-scale randomly generated instances. Our study reveals that the quality of the computed solutions is much closer to optimality than the provable approximation factor.
We consider a constrained energy optimization called Minimum Energy Scheduling Problem (MESP) for a wireless network of N users transmitting over M time slots, where the constraints arise because of interference betwe...
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We consider a constrained energy optimization called Minimum Energy Scheduling Problem (MESP) for a wireless network of N users transmitting over M time slots, where the constraints arise because of interference between wireless nodes that limits their transmission rates along with load and duty-cycle (ON-OFF) restrictions. Since traditional optimization methods using Lagrange multipliers do not work well and are computationally expensive given the nonconvex constraints, we consider approximation schemes for finding the optimal (minimum energy) transmission schedule by discretizing power levels over the interference channel. First, we show the toughness of approximating MESP for an arbitrary number of users N even with a fixed. For any r > 0, we demonstrate that there does not exist any (r, r)-bicriteria approximation for this MESP, unless P = NP. Conversely, we show that there exist good approximations for MESP with given N users transmitting over an arbitrary number of M time slots by developing fully polynomial (1, 1 + epsilon) approximation schemes (FPAS). For any epsilon > 0, we develop an algorithm for computing the optimal number of discrete power levels per time slot (O(1/epsilon)), and use this to design a (1, 1 + epsilon)-FPAS that consumes no more energy than the optimal while violating each rate constraint by at most a 1 + epsilon-factor. For wireless networks with low-cost transmitters, where nodes are restricted to transmitting at a fixed power over active time slots, we develop a two-factor approximation for finding the optimal fixed transmission power value P-opt that results in the minimum energy schedule.
We study the minimal decomposition of octilinear polygons with holes into octilinear triangles and rectangles. This new problem is relevant in the context of modern electronic CAD systems, where the generation and pro...
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We study the minimal decomposition of octilinear polygons with holes into octilinear triangles and rectangles. This new problem is relevant in the context of modern electronic CAD systems, where the generation and propagation of electromagnetic noise into multi layer PCBs has to be detected. It is a generalization of a problem deeply investigated: the minimal decomposition of rectilinear polygons into rectangles. We show that the new problem is NP-hard. We also show the NP-hardness of a related problem, that is the decomposition of an octilinear polygon with holes into octilinear convex polygons. For both problems, we propose efficient approximation algorithms. (C) 2019 Elsevier B.V. All rights reserved.
We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the 2-dimensional plane. Among other results, we give an O(root n)-approximation algorithm for the problem o...
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We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the 2-dimensional plane. Among other results, we give an O(root n)-approximation algorithm for the problem of fi nding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. We give an improved (O) over tilde (n(1/3)) approximation for the case of metrics induced by unweighted trees.
In this paper we study the min-max cycle cover problem with neighborhoods, which is to find a given number of K cycles to collaboratively visit n Points of Interest (POIs) in a 2D space such that the length of the lon...
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In this paper we study the min-max cycle cover problem with neighborhoods, which is to find a given number of K cycles to collaboratively visit n Points of Interest (POIs) in a 2D space such that the length of the longest cycle among the K cycles is minimized. The problem arises from many applications, including employing mobile sinks to collect sensor data in wireless sensor networks (WSNs), dispatching charging vehicles to recharge sensors in rechargeable sensor networks, scheduling Unmanned Aerial Vehicles (UAVs) to monitor disaster areas, etc. For example, consider the application of employing multiple mobile sinks to collect sensor data in WSNs. If some mobile sink has a long data collection tour while the other mobile sinks have short tours, this incurs a long data collection latency of the sensors in the long tour. Existing studies assumed that one vehicle needs to move to the location of a POI to serve it. We however assume that the vehicle is able to serve the POI as long as the vehicle is within the neighborhood area of the POI. One such an example is that a mobile sink in a WSN can receive data from a sensor if it is within the transmission range of the sensor (e.g., within 50 meters). It can be seen that the ignorance of neighborhoods will incur a longer traveling length. On the other hand, most existing studies only took into account the vehicle traveling time but ignore the POI service time. Consequently, although the length of some vehicle tour is short, the total amount of time consumed by a vehicle in the tour is prohibitively long, due to many POIs in the tour. In this paper we first study the min-max cycle cover problem with neighborhoods, by incorporating both neighborhoods and POI service time into consideration. We then propose novel approximation algorithms for the problem, by exploring the combinatorial properties of the problem. We finally evaluate the proposed algorithms via experimental simulations. Experimental results show that the propo
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