We study the following tree search problem: in a given tree T=(V,E) a vertex has been marked and we want to identify it. In order to locate the marked vertex, we can use edge queries. An edge query e asks in which of ...
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We study the following tree search problem: in a given tree T=(V,E) a vertex has been marked and we want to identify it. In order to locate the marked vertex, we can use edge queries. An edge query e asks in which of the two connected components of Ta-e the marked vertex lies. The worst-case scenario where one is interested in minimizing the maximum number of queries is well understood, and linear time algorithms are known for finding an optimal search strategy. Here we study the more involved average-case analysis: A function w:V -> a"e(+) is given which measures the likelihood for a vertex to be the one marked, and we seek to determine the strategy (decision tree) that minimizes the weighted average number of queries. In a companion paper we prove that the above tree search problem is -complete even for the class of trees of bounded diameter or bounded degree. Here, we match this complexity result with a tight algorithmic analysis of the bounded degree instances. We show that any optimal strategy (i.e., one that minimizes the weighted average number of queries) performs at most O(Delta(T)(log|V|+log(w(T)/w (min)))) queries in the worst case, where w(T) is the sum of the likelihoods of the vertices of T, w (min) is the minimum positive likelihood over the vertices of T and Delta(T) is the maximum degree of T. We combine this result with a non-trivial exponential time algorithm to provide an FPTAS for trees with bounded degree. We also show that for unbounded instances a natural greedy strategy attains a 1.62-approximation, improving upon the best known 14-approximation guarantee, previously provided by two of the authors.
This paper studies an extension of the k-median problem under uncertain demand. We are given an n-vertex metric space (V, d) and m client sets {S-i subset of V}(i=1)(m). The goal is to open a set of k facilities F suc...
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This paper studies an extension of the k-median problem under uncertain demand. We are given an n-vertex metric space (V, d) and m client sets {S-i subset of V}(i=1)(m). The goal is to open a set of k facilities F such that the worst-case connection cost over all the client sets is minimized, i.e., min(F subset of V,vertical bar F vertical bar=k) max(i is an element of[m]){Sigma(d(j,f))(j is an element of si)}, where for any F subset of V, d(j, F) = min(f is an element of F) d(j, f). This is a "min-max" or "robust" version of the k-median problem. Note that in contrast to the recent papers on robust and stochastic problems, we have only one stage of decision-making where we select a set of k facilities to open. Once a set of open facilities is fixed, each client in the uncertain client-set connects to the closest open facility. We present a simple, combinatorial O (log n+log m)-approximation algorithm for the robust k-median problem that is based on reweighting/Lagrangean-relaxation ideas. In fact, we give a general framework for (minimization) k-facility location problems where there is a bound on the number of open facilities. We show that if the location problem satisfies a certain "projection" property, then both the robust and stochastic versions of the location problem admit approximation algorithms with logarithmic ratios. We use our framework to give the first approximation algorithms for robust and stochastic versions of several location problems such as k-tree, capacitated k-median, and fault-tolerant k-median.
In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarizati...
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In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarization finds the minimum 'distance' from a reference point, which is usually taken as a vertex of the current outer approximation, to the upper image through a given direction. We propose efficient methods to select the parameters (the reference point and direction vector) of the PS scalarization and analyse the effects of these on the overall performance of the algorithm. Different from the existing vertex selection rules from the literature, the proposed methods do not require solving additional single-objective optimization problems. Using some test problems, we conduct an extensive computational study where three different measures are set as the stopping criteria: the approximation error, the runtime, and the cardinality of the solution set. We observe that the proposed variants have satisfactory results, especially in terms of runtime compared to the existing variants from the literature.
Precomputation of the supported QoS is very important for internet routing. By constructing routing tables before a request arrives, a packet can be forwarded with a simple table lookup. When the QoS information is pr...
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Precomputation of the supported QoS is very important for internet routing. By constructing routing tables before a request arrives, a packet can be forwarded with a simple table lookup. When the QoS information is provided, a node can immediately know whether a certain request can be supported without launching the path finding process. Unfortunately, as the problem of finding a route satisfying two additive constraints is NP-complete, the supported QoS information can only be approximated using a polynomial time mechanism. A good approximation scheme should reduce the error in estimating the actual supported QoS. Nevertheless, existing approaches which determine this error may not truly reflect the performance on admission control, meaning whether a request can be correctly classified as feasible or infeasible. In this paper, we propose using a novel metric, known as distortion area, to evaluate the performance of precomputing the supported QoS. We then analyze the performance of the class of algorithms that approximate the supported QoS through discretizing link metrics. We demonstrate how the performance of these schemes can be enhanced without increasing complexity. Our results serve as a guideline on developing discretization-based approximation algorithms. (C) 2013 The Authors. Published by Elsevier Ltd. All rights reserved.
We consider the problem of computing a large stable matching in a bipartite graph where each vertex ranks its neighbors in an order of preference, perhaps involving ties. Let the matched partner of u in a matching M b...
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We consider the problem of computing a large stable matching in a bipartite graph where each vertex ranks its neighbors in an order of preference, perhaps involving ties. Let the matched partner of u in a matching M be M(u). A matching M is said to be stable if there is no edge (a, b) such that a is unmatched or prefers b to M(a) and similarly, b is unmatched or prefers a to M(b). While a stable matching in G can be easily computed in linear time by the Gale-Shapley algorithm, it is known that computing a maximum size stable matching is APX-hard. In this paper we first consider the case when the preference lists of vertices in A are strict while the preference lists of vertices in B may include ties. This case is also APX-hard and the current best approximation ratio known here is 25/17 which relies on solving an LP. We improve this ratio to 22/15 by a simple linear time algorithm. Here we first compute a half-integral stable matching in and then round it to an integral stable matching M. The ratio is bounded via a payment scheme that charges other components in to cover the costs of length-5 augmenting paths. There will be no length-3 augmenting paths here. We next consider the following special case of two-sided ties, where every tie length is 2. This case is known to be UGC-hard to approximate to within 4/3. We show a 10/7 approximation algorithm here that runs in linear time.
We consider a variety of NP-Complete network connectivity problems. We introduce a novel dual-based approach to approximating network design problems with cut-based linear programming relaxations. This approach gives ...
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We consider a variety of NP-Complete network connectivity problems. We introduce a novel dual-based approach to approximating network design problems with cut-based linear programming relaxations. This approach gives a 3/2-approximation to Minimum 2-Edge-Connected Spanning Subgraph that is equivalent to a previously proposed algorithm. One well-studied branch of network design models ad hoc networks where each node can either operate at high or low power. If we allow unidirectional links, we can formalize this into the problem Dual Power Assignment (DPA). Our dual-based approach gives a 3 / 2-approximation to DPA, improving the previous best approximation known of . Another standard network design problem is Minimum Strongly Connected Spanning Subgraph (MSCS). We propose a new problem generalizing MSCS and DPA called Star Strong Connectivity (SSC). Then we show that our dual-based approach achieves a 1.6-approximation ratio on SSC. As a consequence of our dual-based approximations, we prove new upper bounds on the integrality gaps of these problems.
Recently, Becker and Geiger and Bafna, Berman and Fujito gave 2-approximation algorithms for the feedback vertex set problem in undirected graphs. We show how their algorithms can be explained in terms of the primal-d...
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Recently, Becker and Geiger and Bafna, Berman and Fujito gave 2-approximation algorithms for the feedback vertex set problem in undirected graphs. We show how their algorithms can be explained in terms of the primal-dual method for approximation algorithms, which has been used to derive approximation algorithms for network design problems. In the process, we give a new integer programming formulation for the feedback vertex set problem whose integrality gap is at worst a factor of two;the well-known cycle formulation has an integrality gap of Theta(log n), as shown by Even, Naor, Schieber and Zosin. We also give a new 2-approximation algorithm for the problem which is a simplification of the Bafna et al. algorithm. (C) 1998 Elsevier Science B.V. All rights reserved.
In this paper, we present (1) our implementation of a fast approximation multicommodity flow algorithm, and (2) our implementation of the first provably good approximation algorithm for the minimum normalized cut prob...
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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...
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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.
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