Given a graph G with a set of terminals, two weight functions c and d defined on the edge set of G, and a bound D, a popular NP-hard problem in designing networks is to find the minimum cost Steiner tree (under functi...
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Given a graph G with a set of terminals, two weight functions c and d defined on the edge set of G, and a bound D, a popular NP-hard problem in designing networks is to find the minimum cost Steiner tree (under function c) in G, to connect all terminals in such a way that its diameter (under function d) is bounded by D. Marathe et al. (J. Algoritm. 28(1), 142-171, 1998) proposed an (O(ln n), O(ln n)) approximation algorithm for this bicriteria problem, where n is the number of terminals. The first factor reflects the approximation ratio on the diameter bound D, and the second factor indicates the cost-approximation ratio. Later, Kapoor and Sarwat (Theory Comput. Syst. 41(4), 779-794, 2007) introduced a parameterized approximation algorithm with performance guarantee of (O(p . ln n/ln p), O(ln n/ln p)) for any input value p > 1, by which one can improve the approximation factor for cost at the price of worsening the approximation factor of diameter. In this paper, we consider the reverse scenario in which minimizing the diameter of the solution is more important. We first propose a parameterized approximation algorithm with performance guarantee of (O(ln n/ln p), O(p. H-p . ln n/ln p)), where H-p = 1+ 1/ 2+...+ 1/ p is the pth harmonic number. Parameter p is part of the input and this algorithm runs in polynomial time for constant values of p. We also present another algorithm with approximation ratio of (O(ln n/n p), O(mu . p . ln n/ln p)) which relies on the approximation factor (mu) of the NP-hard problem min-degree constrained minimum spanning tree.
In this paper, we consider the Unsplittable (hard) Capacitated Facility Location Problem (UCFLP) with uniform capacities and present new approximation algorithms for it. This problem is a generalization of the classic...
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In this paper, we consider the Unsplittable (hard) Capacitated Facility Location Problem (UCFLP) with uniform capacities and present new approximation algorithms for it. This problem is a generalization of the classical facility location problem where each facility can serve at most u units of demand and each client must be served by exactly one facility. This problem is motivated by its applications in many practical problems including supply chain problems of indivisible goods (Verter in Foundations of location analysis, chapter 2. International series in operations research and management science. Springer, Berlin, 2011) and the assignment problem in the content distribution networks (Bateni and Hajiaghayi in Proceedings of the nineteenth annual ACM-SIAM symposium on discrete algorithms, pp 805-814, 2009). While there are several approximation algorithms for the soft capacitated version of this problem (in which one can open multiple copies of each facility) or the splittable version (in which the demand of each client can be divided to be served by multiple open facilities), there are very few results for the UCFLP. It is known that it is NP-hard to approximate this problem within any factor without violating the capacities. So we consider bicriteria -approximations where the algorithm returns a solution whose cost is within factor of the optimum and violates the capacity constraints within factor . Shmoys et al. (Proceedings of the twenty-ninth annual ACM symposium on theory of computing, pp 265-274, 1997) were the first to consider this problem and gave a (9, 4)-approximation. Later results imply (O(1), 2)-approximations, however, no constant factor approximation is known with capacity violation of less than 2. We present a framework for designing bicriteria approximation algorithms for this problem and show two new approximation algorithms with factors (9, 3 / 2) and (29.315, 4 / 3). These are the first algorithms with constant approximation in which the viol
This paper provides a new idea for approximating the inventory cost function to be used in a truncated dynamic program for solving the capacitated lot-sizing problem. The proposed method combines dynamic programming w...
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This paper provides a new idea for approximating the inventory cost function to be used in a truncated dynamic program for solving the capacitated lot-sizing problem. The proposed method combines dynamic programming with regression, data fitting, and approximation techniques to estimate the inventory cost function at each stage of the dynamic program. The effectiveness of the proposed method is analyzed on various types of the capacitated lot-sizing problem instances with different cost and capacity characteristics. Computational results show that approximation approaches could significantly decrease the computational time required by the dynamic program and the integer program for solving different types of the capacitated lot-sizing problem instances. Furthermore, in most cases, the proposed approximate dynamic programming approaches can accurately capture the optimal solution of the problem with consistent computational performance over different instances.
We present efficient fixed-parameter and approximation algorithms for the NP-hard problem of computing a maximum agreement forest (MAF) of a pair of multifurcating (nonbinary) rooted trees. Multifurcating trees arise ...
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We present efficient fixed-parameter and approximation algorithms for the NP-hard problem of computing a maximum agreement forest (MAF) of a pair of multifurcating (nonbinary) rooted trees. Multifurcating trees arise naturally as a result of statistical uncertainty in current tree construction methods. The size of an MAF corresponds to the subtree prune-and-regraft distance of the two trees and is intimately connected to their hybridization number. These distance measures are essential tools for understanding reticulate evolution, such as lateral gene transfer, recombination, and hybridization. Our algorithms nearly match the running times of the currently best algorithms for the binary case. This is achieved using a combination of efficient branching rules (similar to but more complex than in the binary case) and a novel edge protection scheme that further reduces the size of the search space the algorithms need to explore.
The bin packing problem is a well-studied problem in combinatorial optimization. In the classical bin packing problem, we are given a list of real numbers in (0, 1] and the goal is to place them in a minimum number of...
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The bin packing problem is a well-studied problem in combinatorial optimization. In the classical bin packing problem, we are given a list of real numbers in (0, 1] and the goal is to place them in a minimum number of bins so that no bin holds numbers summing to more than 1. The problem is extremely important in practice and finds numerous applications in scheduling, routing and resource allocation problems. Theoretically the problem has rich connections with discrepancy theory, iterative methods, entropy rounding and has led to the development of several algorithmic techniques. In this survey we consider approximation and online algorithms for several classical generalizations of bin packing problem such as geometric bin packing, vector bin packing and various other related problems. There is also a vast literature on mathematical models and exact algorithms for bin packing. However, this survey does not address such exact algorithms. In two-dimensional geometric bin packing, we are given a collection of rectangular items to be packed into a minimum number of unit size square bins. This variant has a lot of applications in cutting stock, vehicle loading, pallet packing, memory allocation and several other logistics and robotics related problems. In d-dimensional vector bin packing, each item is a d-dimensional vector that needs to be packed into unit vector bins. This problem is of great significance in resource constrained scheduling and in recent virtual machine placement in cloud computing. We also consider several other generalizations of bin packing such as geometric knapsack, strip packing and other related problems such as vector scheduling, vector covering etc. We survey algorithms for these problems in offline and online setting, and also mention results for several important special cases. We briefly mention related techniques used in the design and analysis of these algorithms. In the end we conclude with a list of open problems. (C) 2016 Elsevier Inc. A
A graph is k-connected if it has k internally-disjoint paths between every pair of nodes. A subset S of nodes in a graph G is a k-connected set if the subgraph G[S] induced by S is k-connected;S is an m-dominating set...
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We revisit the problem of online linear optimization in case the set of feasible actions is accessible through an approximated linear optimization oracle with a factor α multiplicative approximation guarantee. This s...
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The edit distance is a basic string similarity measure used in many applications such as text mining, signal processing, bioinformatics, and so on. However, the computational cost can be a problem when we repeat many ...
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Sequence classification algorithms, such as SVM, require a definition of distance (similarity) measure between two sequences. A commonly used notion of similarity is the number of matches between k-mers (k-length subs...
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We consider the distributed version of the Multiple Knapsack Problem (MKP), where m items are to be distributed amongst n processors, each with a knapsack. We propose different distributed approximation algorithms wit...
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