Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals...
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Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of genes on corresponding chromosomes, and a reversal takes an arbitrary substring of elements, and reverses their order. For this problem, we develop two algorithms: a greedy approximation algorithm, that finds a solution provably close to optimal in O(n(2)) time and O(n) space for n-element permutations, and a branch-and-bound exact algorithm, that finds an optimal solution in O(mL(n, n)) time and O(n(2)) space, where m is the size of the branch-and-bound search tree, and L(n, n) is the time tc, solve a linear program of n variables and n constraints. The greedy algorithm is the first to come within a constant factor of the optimum;it guarantees a solution that uses no more than twice the minimum number of reversals. The lower and upper bounds of the branch-and-bound algorithm are a novel application of maximum-weight matchings, shortest paths, and linear programming. In a series of experiments, we study the performance of an implementation on random permutations, and permutations generated by random reversals. For permutations differing by k random reversals, we find that the average upper bound on reversal distance estimates k to within one reversal for k < 1/2n and n less than or equal to 100. For the difficult case of random permutations, we find that the average difference between the upper and lower bounds is less than three reversals for n less than or equal to 50. Due to the tightness of these bounds, we can solve, to optimality, problems on 30 elements in a few minutes of computer time. This approaches the scale of mitochondrial genomes.
We consider the frequency assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with a certain geometric structure. The problem of broadcast scheduling can ...
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We consider the frequency assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with a certain geometric structure. The problem of broadcast scheduling can be cast as a variant of the vertex coloring problem (called the distance-2 coloring problem) on the graph that models a given packet radio network, We present efficient approximation algorithms for the distance-2 coloring problem for various geometric graphs including those that naturally model a large class of packet radio networks. The class of graphs considered include (r, s)-civilized graphs, planar graphs, graphs with bounded genus, etc.
Betweenness centrality, which measures the contribution of an individual node to the network's connectivity by counting the number of shortest paths a node appears in, is widely used for the analysis of the comple...
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Betweenness centrality, which measures the contribution of an individual node to the network's connectivity by counting the number of shortest paths a node appears in, is widely used for the analysis of the complex networks. The computation of exact betweenness centrality is prohibitively expensive for large networks, given a worst-case complexity of O(N * E), where N is the number of nodes and E is the number of edges in the network. Accordingly, a multitude of approximation algorithms has been proposed in the literature. Obtaining an overview of the state of the art is difficult, given a combination of numerous algorithms, parameters, and network topologies. In this paper, we report on the results of the probably largest benchmark performed in this field. Specifically, we select 100 networks with distinct topologies and scales, covering various domains. We devise and compare eight selected measures to evaluate the accuracy of the approximation, compared with the exact betweenness computation. All experiments, including those to obtain the exact betweenness values, have been performed on one computer using a single thread, in order to provide a fair comparison. We implemented typical approximation methods and report sensitivity analysis results with a variety of parameters. We find that a uniformly random sampling method, one of the earliest proposed methods in this field, still delivers the best performance, nicely addressing a sweet spot between quality and runtime complexity. In addition, we carried out robustness experiments based on the ranking order of approximated betweenness, in order to show the effect of different approximations on a real-world task. Our study aims at being a reference for choosing a betweenness approximation method, with consideration of network type, the required level of accuracy, and available computational resources.
Mathematical research on multicriteria optimization problems predominantly revolves around the set of Pareto optimal solutions. In practice, on the other hand, methods that output a single solution are more widespread...
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Mathematical research on multicriteria optimization problems predominantly revolves around the set of Pareto optimal solutions. In practice, on the other hand, methods that output a single solution are more widespread. Reference point methods are a successful example of this approach and are widely used in real-world multicriteria optimization. A reference point solution is the solution closest to a given reference point in the objective space. We study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets. In particular, we establish that, in terms of computational complexity, computing approximate reference point solutions is polynomially equivalent to approximating the Pareto set. Complementing these results, we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization. In particular, we lift the link between dynamic programming and FPTAS, as well as certain LP-rounding techniques. The latter applies, e.g., to SET COVER and several machine scheduling problems. (C) 2016 Elsevier B.V. All rights reserved.
We consider approval voting elections in which each voter votes for a (possibly empty) set of candidates and the outcome consists of a set of k candidates for some parameter k, e.g., committee elections. We are intere...
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In the stochastic orienteering problem, we are given a finite metric space, where each node contains a job with some deterministic reward and a random processing time. The processing time distributions are known and i...
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In the stochastic orienteering problem, we are given a finite metric space, where each node contains a job with some deterministic reward and a random processing time. The processing time distributions are known and independent across nodes. However the actual processing time of a job is not known until it is completely processed. The objective is to compute a nonanticipatory policy to visit nodes (and run the corresponding jobs) so as to maximize the total expected reward, subject to the total distance traveled plus the total processing time being at most a given budget of B. This problem combines aspects of the stochastic knapsack problem with uncertain item sizes as well as the deterministic orienteering problem. In this paper, we consider both nonadaptive and adaptive policies for Stochastic Orienteering. We present a constant-factor approximation algorithm for the nonadaptive version and an O(log log B)-approximation algorithm for the adaptive version. We extend both these results to directed metrics and a more general sequence orienteering problem. Finally, we address the stochastic orienteering problem when the node rewards are also random and possibly correlated with the processing time and obtain an O(log n log B)-approximation algorithm;here n is the number of nodes in the metric. All our results for adaptive policies also bound the corresponding "adaptivity gaps".
Let c, k be two positive integers. Given a graph , the c-Load Coloring problem asks whether there is a c-coloring such that for every , there are at least k edges with both endvertices colored i. Gutin and Jones (Inf ...
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Let c, k be two positive integers. Given a graph , the c-Load Coloring problem asks whether there is a c-coloring such that for every , there are at least k edges with both endvertices colored i. Gutin and Jones (Inf Process Lett 114:446-449, 2014) studied this problem with . They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of , we obtain a kernel with less than 4k vertices and less than edges. These results imply that for any fixed , c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.
In this paper we present an n(O(k1-1/d))-time algorithm for solving the k-center problem in R-d, under L-infinity- and L-2-metrics. The algorithm extends to other metrics, and to the discrete k-center problem. We also...
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In this paper we present an n(O(k1-1/d))-time algorithm for solving the k-center problem in R-d, under L-infinity- and L-2-metrics. The algorithm extends to other metrics, and to the discrete k-center problem. We also describe a simple (1 + epsilon)-approximation algorithm for the k-center problem, with running time O(n log k) + (k/epsilon)(O(k1-1/d)). Finally, we present an n(O(k1-1/d))-time algorithm for solving the L-capacitated k-center problem, provided that L = Omega(n/k(1-1/d)) or L = O(I).
The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2 - 1 / (n - 1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n(...
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The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2 - 1 / (n - 1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n(3) log n) time-it applies the primaldual scheme once for each of the n vertices of the graph. We present a primal-dual algorithm that runs in O(n(2) log n), as it applies this scheme only once, and achieves the slightly better ratio of (2 - 2/n). We also show a tight example for the analysis of the algorithm and discuss briefly a couple of other algorithms described in the literature. (c) 2007 Elsevier B.V. All rights reserved.
We introduce a variant of the Shortest Path Problem (SPP), in which we impose additional constraints on the acceleration over the arcs, and call it Bounded Acceleration SPP (BASP). This variant is inspired by an indus...
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We introduce a variant of the Shortest Path Problem (SPP), in which we impose additional constraints on the acceleration over the arcs, and call it Bounded Acceleration SPP (BASP). This variant is inspired by an industrial application: a vehicle needs to travel from its current position to a target one in minimum-time, following pre-defined geometric paths connecting positions within a facility, while satisfying some speed and acceleration constraints depending on the vehicle position along the currently traveled path. We characterize the complexity of BASP, proving its NP-hardness. We also show that, under additional hypotheses on problem data, the problem admits a pseudo-polynomial time-complexity algorithm. Moreover, we present an approximation algorithm with polynomial time-complexity with respect to the data of the original problem and the inverse of the approximation factor e. Finally, we present some computational experiments to evaluate the performance of the proposed approximation algorithm.
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