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.
We consider the interval constrained coloring problem, which appears in the interpretation of experimental data in biochemistry. Monitoring hydrogen-deuterium exchange rates via mass spectroscopy experiments is a meth...
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We consider the interval constrained coloring problem, which appears in the interpretation of experimental data in biochemistry. Monitoring hydrogen-deuterium exchange rates via mass spectroscopy experiments is a method used to obtain information about protein tertiary structure. The output of these experiments provides data about the exchange rate of residues in overlapping segments of the protein backbone. These segments must be re-assembled in order to obtain a global picture of the protein structure. The interval constrained coloring problem is the mathematical abstraction of this re-assembly process. The objective of the interval constrained coloring problem is to assign a color (exchange rate) to a set of integers (protein residues) such that a set of constraints is satisfied. Each constraint is made up of a closed interval (protein segment) and requirements on the number of elements that belong to each color class (exchange rates observed in the experiments). We show that the problem is NP-complete for arbitrary number of colors and we provide algorithms that given a feasible instance find a coloring that satisfies all the coloring requirements within +/- 1 of the prescribed value. In light of our first result, this is essentially the best one can hope for. Our approach is based on polyhedral theory and randomized rounding techniques. Furthermore, we consider a variant of the problem where we are asked to find a coloring satisfying as many fragments as possible. If we relax the coloring requirements by a small factor of (1+epsilon), we propose an algorithm that finds a coloring "satisfying" this maximum number of fragments and that runs in quasi-polynomial time if the number of colors is polylogarithmic.
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 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.
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.
In this paper we investigate the problem of approximating the fraction of truth assignments that satisfy a Boolean formula with some restricted form of DNF under distributions with limited independence between random ...
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In this paper we investigate the problem of approximating the fraction of truth assignments that satisfy a Boolean formula with some restricted form of DNF under distributions with limited independence between random variables. Let F be a DNF formula on n variables with m clauses in which each literal appears at most once. We prove that if D is [k log m]-wise independent, then \Pr-D[F]-Pr-U[F]\ less than or equal to (1/2)(Omega(k)), where U denotes the uniform distribution and Pr-D[F] denotes the probability that F is satisfied by a truth assignment chosen according to distribution D (similarly for Pr-U[F]). Using the result, we also derive the following: For formulas satisfying the restriction described above and for any constant c, there exists a probability distribution D, with size polynomial in log n and m, such that \Pr-D[F]-Pr-U[F]\ less than or equal to c holds.
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
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.
The problem of covering edges and vertices in a graph (or in a hypergraph) was motivated by a problem arising in the context of the component assembly problem. The problem is as follows: given a graph and a clique siz...
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The problem of covering edges and vertices in a graph (or in a hypergraph) was motivated by a problem arising in the context of the component assembly problem. The problem is as follows: given a graph and a clique size Ic, find the minimum number of k-cliques such that all edges and vertices of the graph are covered by (included in) the cliques. This paper provides a collection of approximation algorithms for various clique sizes with proven worst-case bounds. The problem has a natural extension to hypergraphs, for which we consider one particular class. The k-clique covering problem can be formulated as a set covering problem. It is shown that the algorithms we design, which exploit the structure of this special set covering problem, have better performance than those derived from direct applications of general purpose algorithms for the set covering. In particular, these special classes of set covering problems can be solved with better worst-case bounds and/or complexity than if treated as general set covering problems.
A general parallel task scheduling problem is considered. A task can be processed in parallel on one of several alternative subsets of processors. The processing time of the task depends on the subset of processors as...
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A general parallel task scheduling problem is considered. A task can be processed in parallel on one of several alternative subsets of processors. The processing time of the task depends on the subset of processors assigned to the task. We first show the hardness of approximating the problem for both preemptive and nonpreemptive cases in the general setting. Next we focus on linear array network of in processors. We give an approximation algorithm of ratio O(log m) for nonpreemptive scheduling, and another algorithm of ratio 2 for preemptive scheduling. Finally, we give a nonpreemptive scheduling algorithm of ratio O(log(2) m) for m x m in two-dimensional meshes. (C) 2002 Elsevier Science B.V. All rights reserved.
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