We describe approximation algorithms with bounded performance guarantees for the following problem: A graph is given with edge weights satisfying the triangle inequality, together with two numbers k and p. Find k disj...
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We describe approximation algorithms with bounded performance guarantees for the following problem: A graph is given with edge weights satisfying the triangle inequality, together with two numbers k and p. Find k disjoint subsets of p vertices each, so that the total weight of edges within subsets is maximized. (C) 1997 Elsevier Science B.V.
We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first ri...
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We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most , throughout the uniqueness region of the Gibbs measure on the infinite -regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.
In this article we present approximation algorithms for the Arc Orienteering Problem (AOP). We propose a polylogarithmic approximation algorithm in directed graphs, while in undirected graphs we give a (6 + epsilon + ...
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In this article we present approximation algorithms for the Arc Orienteering Problem (AOP). We propose a polylogarithmic approximation algorithm in directed graphs, while in undirected graphs we give a (6 + epsilon + o(1)) and a (4 + epsilon)-approximation algorithm for arbitrary instances and instances of unit profit, respectively. Also, an inapproximability result for the AOP is obtained as well as approximation algorithms for the Mixed Orienteering Problem. (C) 2014 Elsevier B.V. All rights reserved.
We introduce the class cover problem, a variant of disk cover with forbidden regions, with applications to classification and facility location problems. We prove similar hardness results to disk cover. We then presen...
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We introduce the class cover problem, a variant of disk cover with forbidden regions, with applications to classification and facility location problems. We prove similar hardness results to disk cover. We then present a polynomial-time approximation algorithm for class cover that performs within a ln n + 1 factor of optimal, which is nearly tight under standard hardness assumptions. In the special case that the points lie in a d-dimensional space with Euclidean norm, for some fixed constant d, we obtain a polynomial time approximation scheme.
The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set...
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Delta) + 2 and H(Delta) + 2 are presented, where Delta is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (c(n) + 1) ln n where c(n) ln k is the approximation factor for the node weighted Steiner tree problem (currently c(n) = 1.6103). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c + 1)H(Delta) + c - 1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644).
We study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in [T. Lengauer, J. Assoc. Comput. Mach., 36(1989), pp. 474-509] or are specifi...
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We study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in [T. Lengauer, J. Assoc. Comput. Mach., 36(1989), pp. 474-509] or are specified by one-dimensional finite narrow periodic specifications as in [E. Wanke, Paths and cycles in finite periodic graphs, in Lecture Notes in Comp. Sci. 711, Springer-Verlag, New York, 1993, pp. 751-760]. We show that, for most of the problems Pi considered when specified using k-level-restricted hierarchical specifications or k-narrow periodic specifications, the following hold. (i) Let p be any performance guarantee of a polynomial time approximation algorithm for Pi, when instances are specified using standard specifications. Then For All epsilon > 0, Pi has a polynomial time approximation algorithm with performance guarantee (1 + epsilon)p. (ii) Pi has a polynomial time approximation scheme when restricted to planar instances. These are the first polynomial time approximation schemes for PSPACE-hard hierarchically or periodically specified problems. Since several of the problems considered are PSPACE-hard, our results provide the first examples of natural PSPACE-hard optimization problems that have polynomial time approximation schemes. This answers an open question in Condon et al. [Chicago J. Theoret. Comput. Sci., 1995, Article 4].
This paper presents approximation algorithms for two extensions of the set cover problem: a graph-based extension known as the Max-Rep or Label-Cover(MAX)problem, and a color-based extension known as the Red-Blue Set ...
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This paper presents approximation algorithms for two extensions of the set cover problem: a graph-based extension known as the Max-Rep or Label-Cover(MAX)problem, and a color-based extension known as the Red-Blue Set Cover problem. First, a randomized algorithm guaranteeing approximation ratio root n with high probability is proposed for the Max-Rep (or Label-Cover(MAX)) problem, where n is the number of vertices in the graph. This algorithm is then generalized into a 4 root n-ratio algorithm for the nonuniform version of the problem. Secondly, it is shown that the Red-Blue Set Cover problem can be approximated with ratio 2 root n log beta, where n is the number of sets and beta is the number of blue elements. Both algorithms can be adapted to the weighted variants of the respective problems, yielding the same approximation ratios. (C) 2006 Elsevier B.V. All rights reserved.
Stochastic optimization has established itself as a major method to handle uncertainty in various optimization problems by modeling the uncertainty by a probability distribution over possible ***,the main focus in st...
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Stochastic optimization has established itself as a major method to handle uncertainty in various optimization problems by modeling the uncertainty by a probability distribution over possible ***,the main focus in stochastic optimization has been various stochastic mathematical programming(such as linear programming,convex programming).In recent years,there has been a surge of interest in stochastic combinatorial optimization problems from the theoretical computer science *** this article,we survey some of the recent results on various stochastic versions of classical combinatorial optimization *** most problems in this domain are NP-hard(or#P-hard,or even PSPACE-hard),we focus on the results which provide polynomial time approximation algorithms with provable approximation *** discussions are centered around a few representative problems,such as stochastic knapsack,stochastic matching,multi-armed bandit *** use these examples to introduce several popular stochastic models,such as the fixed-set model,2-stage stochastic optimization model,stochastic adaptive probing model etc,as well as some useful techniques for designing approximation algorithms for stochastic combinatorial optimization problems,including the linear programming relaxation approach,boosted sampling,content resolution schemes,Poisson approximation *** also provide some open research questions along the *** purpose is to provide readers a quick glimpse to the models,problems,and techniques in this area,and hopefully inspire new contributions.
We study the load-balanced capacitated vehicle routing problem (LBCVRP): the problem is to design a collection of tours for a fixed fleet of vehicles with capacityQto distribute a supply from a single depot between a ...
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We study the load-balanced capacitated vehicle routing problem (LBCVRP): the problem is to design a collection of tours for a fixed fleet of vehicles with capacityQto distribute a supply from a single depot between a number of predefined clients, in a way that the total traveling cost is a minimum, and the vehicle loads are balanced. The unbalanced loads cause the decrease of distribution quality especially in business environments and flexibility in the logistics activities. The problem being NP-hard, we propose two approximation algorithms. When the demands are equal, we present a((1-1Q)rho+3/2)-approximation algorithm that finds balanced loads. Here, rho is the approximation ratio for the known metric traveling salesman problem (TSP). This result leads to a 2.5-1/Q approximation ratio for the tree metrics since an optimal solution can be found for the TSP on a tree. We present an improved2- approximation algorithm. When the demands are unequal, we focus on obtaining approximate solutions since finding balanced loads is NP-complete. We propose an algorithm that provides a4-approximation for the balance of the loads. We assume a second approach to get around the difficulties of the feasibility. In this approach, we redefine and convert the problem into a multi-objective problem. The algorithm we propose has a 4 factor of approximation.
We consider a purchase/inventory control problem in which the purchase price and demand are stochastic, a common situation encountered by firms that replenish in a foreign currency or from commodity markets. More spec...
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We consider a purchase/inventory control problem in which the purchase price and demand are stochastic, a common situation encountered by firms that replenish in a foreign currency or from commodity markets. More specifically, we assume that the demand follows a Poisson arrival process and that the log-price evolves according to a general Wiener process. Under these circumstances, the optimal policy is a state dependent base-stock policy that can be described as a series of threshold prices. An iterative procedure for determining the optimal thresholds has been derived earlier but, even for the simplest price process, the solution quickly becomes numerically intractable. To deal with this, we propose an approximation that allows us to derive simple heuristics for finding thresholds that are close to optimal. For certain price processes the heuristics are just a series of closed-form expressions. The computational complexity is reduced significantly, and the numerical study shows that the new heuristics perform considerably better than earlier suggested heuristics.
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