We introduce and study a generalization of the classic sequential testing problem, asking to identify the correct state of a given series system that consists of independent stochastic components. In this setting, cos...
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We introduce and study a generalization of the classic sequential testing problem, asking to identify the correct state of a given series system that consists of independent stochastic components. In this setting, costly tests are required to examine the state of individual components, which are sequentially tested until the correct system state can be uniquely identified. The goal is to propose a policy that minimizes the expected testing cost, given a-priori probabilistic information on the stochastic nature of each individual component. Unlike the classic setting, where variables are tested one after the other, we allow multiple tests to be conducted simultaneously, at the expense of incurring an additional set-up cost. The main contribution of this article consists in showing that the batch testing problem can be approximated in polynomial time within factor 6.829 + epsilon, for any fixed epsilon epsilon (0, 1). In addition, we explain how, in spite of its highly nonlinear objective function, the batch testing problem can be formulated as an approximate integer program of polynomial size, while blowing up its expected cost by a factor of at most 1 + epsilon. Finally, we conduct extensive computational experiments, to demonstrate the practical effectiveness of these algorithms as well as to evaluate their limitations. (C) 2016 Wiley Periodicals, Inc.
Betweenness centrality quantifies the importance of graph nodes in a variety of applications including social, biological and communication networks. Its computation is very costly for large graphs;therefore, many app...
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ISBN:
(纸本)9781450352826
Betweenness centrality quantifies the importance of graph nodes in a variety of applications including social, biological and communication networks. Its computation is very costly for large graphs;therefore, many approximate methods have been proposed. Given the lack of a golden standard, the accuracy of most approximate methods is evaluated on tiny graphs and is not guaranteed to be representative of realistic datasets that are orders of magnitude larger. In this paper, we develop BeBeCA, a benchmark for betweenness centrality approximation methods on large graphs. Specifically: (i) We generate a golden standard by deploying a parallel implementation of Brandes algorithm using 96,000 CPU cores on a supercomputer to compute exact betweenness centrality values for several large graphs with up to 126M edges. (ii) We propose an evaluation methodology to assess various aspects of approximation accuracy, such as average error and quality of node ranking. (iii) We survey a large number of existing approximation methods and compare their performance and accuracy using our benchmark. (iv) We publicly share our benchmark, which includes the golden standard exact betweenness centrality values together with the scripts that implement our evaluation methodology;for researchers to compare their own algorithms and practitioners to select the appropriate algorithm for their application and data.
We present a nested local search algorithm to approximate several variants of metric two-stage stochastic facility location problems. These problems are generalizations of the well-studied metric uncapacitated facilit...
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ISBN:
(数字)9783319517414
ISBN:
(纸本)9783319517414;9783319517407
We present a nested local search algorithm to approximate several variants of metric two-stage stochastic facility location problems. These problems are generalizations of the well-studied metric uncapacitated facility location problem, taking uncertainties in demand values and costs into account. The proposed nested local search procedure uses three facility operations: adding, dropping, and swapping. To the best of our knowledge, this is the first constant-factor local search approximation for two-stage stochastic facility location problems. Besides traditional direct assignments from clients to facilities, we also investigate shared connections via capacitated trees and tours. We obtain the first constant-factor approximation algorithms for both connection types in the setting of two-stage stochastic optimization. Our algorithms admit order-preserving metrics and thus significantly generalize and improve the allowed mutability of the metric in comparison to previous algorithms, which only allow scenario-dependent inflation factors.
The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the sm...
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The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small-Set Expansion problem, which, for a parameter delta e (0,1/2], asks to compute the cut having the least expansion while having at most delta fraction of the vertices on the smaller side of the cut. We present two algorithms. Our first algorithm gives an O similar to(delta(-) (1) root logn)-approximation. The second algorithm finds a set with expansion O-similar to(delta(-1)(root d(max)r(-1)logr Phi* + Phi*)) in an r-uniform hypergraph with maximum degree d(max) (where Phi* is the expansion of the optimal solution). Using these results, we also obtain algorithms for the Small-Set Vertex Expansion problem: we get an O similar to(sigma(-1)root logn)-approximation algorithm and an algorithm that finds a set with vertex expansion. O-similar to(dagger(-1)root Phi(V)logd(max )+ delta(-1)Phi(V)) (where Phi(V) is the vertex expansion of the optimal solution). For delta = 1/2, Hypergraph Small-Set Expansion is equivalent to the hypergraph expansion problem. In this case, our approximation factor of O(root logn) for expansion in hypergraphs matches the corresponding approximation factor for expansion in graphs due to Arora, Rao, and Vazirani (JACM 2009).
We consider the following two deterministic inventory optimization problems with non-stationary demands. Submodular joint replenishment problem. This involves multiple item types and a single retailer who faces demand...
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We consider the following two deterministic inventory optimization problems with non-stationary demands. Submodular joint replenishment problem. This involves multiple item types and a single retailer who faces demands over a finite planning horizon of T periods. In each time period, any subset of item-types can be ordered incurring a joint ordering cost which is submodular. Moreover, items can be held in inventory while incurring a holding cost. The objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs. Inventory routing problem. This involves a single depot that stocks items, and multiple retailer locations facing demands over a finite planning horizon of T periods. In each time period, any subset of locations can be visited using a vehicle originating from the depot. There is also cost incurred for holding items at any retailer. The objective here is to satisfy all demands while minimizing the sum of routing and holding costs. We present a unified approach that yields -factor approximation algorithms for both problems when the holding costs are polynomial functions. A special case is the classic linear holding cost model, wherein this is the first sub-logarithmic approximation ratio for either problem.
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.
Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G, there is a vertex u in D ...
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Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G, there is a vertex u in D such that the distance between v and u in G is at most k. We study the problem of finding a minimum k-hop connected dominating set of a graph (Mink-CDS). We prove that Mink-CDS is NP-hard on planar bipartite graphs of maximum degree 4. We also prove that Mink-CDS is APX-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for Mink-CDS on bipar- tite and on (1, 2)-split graphs. Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices. We also discuss the complex- ity of computing this graph parameter. On the positive side, we show an approximation algorithm for Mink-CDS. When k = 1, we present two new approximation algorithms for the weighted version of the problem, one of them restricted to graphs with a poly- nomially bounded number of minimal separators. Finally, also for the weighted variant of the problem where k = 1, we discuss an integer linear programming formulation and conduct a polyhedral study of its associated polytope.
In the Maximum Duo-Preservation String Mapping problem we are given two strings and wish to map the letters of the former to the letters of the latter as to maximise the number of duos. A duo is a pair of consecutive ...
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A k-dominating set in a graph G = (V,E) is a set U ¢ V such that ever vertex of G is either in U or has at least k neighbors in U. In this paper we give simple distributed approximation algorithms in the local mo...
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A k-dominating set in a graph G = (V,E) is a set U ¢ V such that ever vertex of G is either in U or has at least k neighbors in U. In this paper we give simple distributed approximation algorithms in the local model for the minimum k-dominating set problem for k ≥ 2 in graphs with no K3,h-minor and graphs with no K4,4-minor. In particular, this gives fast distributed approximations for graphs of bounded genus and linklessly embeddable graphs. The algorithms give a constant approximation ratio and run in a constant number of rounds. In addition, we will give a (1 + €)-approximation for an arbitrary fixed € > 0 which runs in O(log∗ n) rounds where n is the order of a graph.
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.
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