X-BalancedCC multiwinner voting rules constitute an attractive but computationally intractable compromise between the proportionality provided by the Monroe rule and the diversity provided by the Chamberlin-Courant ru...
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ISBN:
(纸本)9781450363099
X-BalancedCC multiwinner voting rules constitute an attractive but computationally intractable compromise between the proportionality provided by the Monroe rule and the diversity provided by the Chamberlin-Courant rule. We show how to use the Greedy-Monroe algorithm to get improved approximation results for the X-BalancedCC rules and for the Chamberlin-Courant rule, by appropriately setting a " schedule" for the sizes of virtual districts. We describe a polynomial-time algorithm for computing a schedule that guarantees high approximation ratio, but show that finding the best possible schedule for a given election is NP-hard. We further evaluate our algorithms experimentally and show that they perform very well in practice.
We study the Edge-Disjoint Paths with Congestion (EDPwC) problem in undirected networks in which we must integrally route a set of demands without causing large congestion on an edge. We present a (polylog(n), poly(lo...
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ISBN:
(纸本)9780769542447
We study the Edge-Disjoint Paths with Congestion (EDPwC) problem in undirected networks in which we must integrally route a set of demands without causing large congestion on an edge. We present a (polylog(n), poly(log log n))-approximation, which means that if there exists a solution that routes X demands integrally on edge-disjoint paths (i.e. with congestion 1), then the approximation algorithm can route X / polylog(n) demands with congestion poly(log log n). The best previous result for this problem was a (n(1/beta), beta)-approximation for beta < log n.
This paper presents approximation algorithms for two problems. First, a randomized algorithm guaranteeing approximation ratio rootn with high probability is proposed for the Max-Rep problem of [Kor98], or the Label-Co...
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ISBN:
(纸本)3540676902
This paper presents approximation algorithms for two problems. First, a randomized algorithm guaranteeing approximation ratio rootn with high probability is proposed for the Max-Rep problem of [Kor98], or the Label-Cover(MAX) problem (cf. [Hoc95]), where n is the number of vertices in the graph. This algorithm is then generalized into a 4 rootn-ratio algorithm for the nonuniform version of the problem. Secondly, it is shown that the Red-Blue Set Cover problem of [CDKM00] can be approximated with ratio 2 root nlog 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.
We consider optimization problems that can be formulated as minimizing the cost of a feasible solution w(T)x over an arbitrary combinatorial feasible set F subset of {0,1}(n). For these problems we describe a broad cl...
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ISBN:
(纸本)9783642153686
We consider optimization problems that can be formulated as minimizing the cost of a feasible solution w(T)x over an arbitrary combinatorial feasible set F subset of {0,1}(n). For these problems we describe a broad class of corresponding stochastic problems where the cost vector W has independent random components, unknown at the time of solution. A natural and important objective that incorporates risk in this stochastic setting is to look for a feasible solution whose stochastic cost has a small tail or a small convex combination of mean and standard deviation. Our models can be equivalently reformulated as nonconvex programs for which no efficient algorithms are known. In this paper, we make progress on these hard problems. Our results are several efficient general-purpose approximation schemes. They use as a black-box (exact or approximate) the solution to the underlying deterministic problem and thus immediately apply to arbitrary combinatorial problems. For example, from an available delta-approximation algorithm to the linear problem, we construct a delta(1 + epsilon)-approximation algorithm for the stochastic problem, which invokes the linear algorithm only a logarithmic number of times in the problem input (and polynomial in 1/epsilon), for any desired accuracy level epsilon > 0. The algorithms are based on a geometric analysis of the curvature and approximability of the nonlinear level sets of the objective functions.
Motivated by applications in grid computing and project management;we study multiprocessor scheduling in scenarios where there is uncertainty in the successful execution of jobs when assigned to processors. We conside...
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ISBN:
(纸本)9781595936677
Motivated by applications in grid computing and project management;we study multiprocessor scheduling in scenarios where there is uncertainty in the successful execution of jobs when assigned to processors. We consider the problem of multiprocessor scheduling under uncertainty, in which we are given n unit-time jobs and m machines, a directed acyclic graph C giving the dependencies among the jobs, and for every job j and machine i, the probability p(ij) of the successful completion of job j when scheduled on machine i in any given particular step. The goal of the problem is to find a schedule that minimizes the expected makespan, that is, the expected completion time of all the jobs. The problem of multiprocessor scheduling under uncertainty was introduced by Malewicz and was shown to be NP-hard even when all the jobs are independent. In this paper, we present polynomial-time approximation algorithms for the problem;for special cases of the dag C. We obtain an O(log n)-approximation for the case of independent jobs, an O(log m log n log(n + m)/log log(n + m))-approximation when C is a collection of disjoint chains, an O(log m log(2) n)-approximation when C is a collection of directed out- or in-trees, and an O(log m log(2) n log(n + m)/log log(n + m))-approximation when C is a directed forest.
The computational complexity of combinatorial multiple objective programming problems is investigated. NP-completeness and #P-completeness results are presented. Using two definitions of approximability, general resul...
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We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that min...
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ISBN:
(纸本)9783642153686
We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(logn/log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on the recent result of Asadpour, Goemans, Madry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi.
We consider the dynamic map labeling problem: given a set of rectangular labels on the map, the goal is to appropriately select visible ranges for all the labels such that no two consistent labels overlap at every sca...
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ISBN:
(纸本)9783319080161;9783319080154
We consider the dynamic map labeling problem: given a set of rectangular labels on the map, the goal is to appropriately select visible ranges for all the labels such that no two consistent labels overlap at every scale and the sum of total visible ranges is maximized. We propose approximation algorithms for several variants of this problem. For the simple ARO problem, we provide a 3c log n-approximation algorithm for the unit-width rectangular labels if there is a c-approximation algorithm for unit-width label placement problem in the plane;and a randomized polynomial-time O(log n log log n)-approximation algorithm for arbitrary rectangular labels. For the general ARO problem, we prove that it is NP-complete even for congruent square labels with equal selectable scale range. Moreover, we contribute 12-approximation algorithms for both arbitrary square labels and unit-width rectangular labels, and a 6-approximation algorithm for congruent square labels.
We study the problem of finding the optimal code of size k for a given classical-quantum channel, with input space X and output space of dimension d, from an algorithmic point of view. We show that unlike the classica...
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ISBN:
(纸本)9781538692912
We study the problem of finding the optimal code of size k for a given classical-quantum channel, with input space X and output space of dimension d, from an algorithmic point of view. We show that unlike the classical case, the relevant function is not submodular, and does not have the diminishing returns property. We also study a semidefinite programming relaxation, which corresponds to the setting where the sender and receiver have a non-signalling box, and show that it can be rounded in two different ways. If the SDP gives a value of p for the success probability, then a simple rounding strategy returns a code of size k with success probability at least p/O(log vertical bar X vertical bar log d). The second rounding method is based on the pretty good measurement and returns a code with success probability close to p, but the code is smaller: it has size k center dot Omega(p(2)).
In this paper, we introduce the study of prize-collecting network design problems having general connectivity requirements. Prior work considered only 0-1 or very limited connectivity requirements. We introduce genera...
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ISBN:
(纸本)9783540939795
In this paper, we introduce the study of prize-collecting network design problems having general connectivity requirements. Prior work considered only 0-1 or very limited connectivity requirements. We introduce general connectivity requirements in the prize-collecting generalized Steiner tree framework of Hajiaghayi and Jain [9], and consider penalty functions linear in the violation of the connectivity requirements. Using Jain's iterated rounding algorithm [11] as a black box, and ideas from Goemans [7] and Levi, Lodi, Sviridenko [14], we give a 2.54-factor approximation algorithm for the problem. We also generalize the 0-1 requirements of PCF problem introduced by Sharma, Swamy, and Williamson [15] to include general connectivity requirements. Here we assume that the monotone submodular penalty function of Sharma et al. is generalized to a multiset function that can be decomposed into functions in the same form as that of Sharma et al. Using ideas from Goemans and Berstimas [6], we give an (a log K)-approximation algorithm for the resulting problem, where K is the maximum connectivity requirement, and alpha = 2.54.
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