The metric dimension of a graph G is the smallest size of a set R of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in R. Computing the metric dimension is NP-hard, ev...
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The metric dimension of a graph G is the smallest size of a set R of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in R. Computing the metric dimension is NP-hard, even when restricting inputs to bipartite graphs. We present a linear-time algorithm for computing the metric dimension for chain graphs, which are bipartite graphs whose vertices can be ordered by neighborhood inclusion. (C) 2015 Elsevier B.V. All rights reserved.
Given a digraph D = (V, A) and a positive integer k, an arc set F C A is called a k-arborescence if it is the disjoint union of k spanning arborescences. The problem of finding a minimum cost k-arborescence is known t...
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ISBN:
(纸本)9781510819672
Given a digraph D = (V, A) and a positive integer k, an arc set F C A is called a k-arborescence if it is the disjoint union of k spanning arborescences. The problem of finding a minimum cost k-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost k-arborescence. For k = 1. the problem was solved in [A. Bernath, G. Pap, Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm for general k that has polynomial running time if k is fixed.
Three men, each with a sister, must cross a river using a boat that can carry only two people in such a way that a sister is never left in the company of another man if her brother is not present. This very famous pro...
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Three men, each with a sister, must cross a river using a boat that can carry only two people in such a way that a sister is never left in the company of another man if her brother is not present. This very famous problem appeared in the Latin book "Problems to Sharpen the Young," one of the earliest collections of recreational mathematics. This paper considers a generalization of such "river crossing problems" and provides a new formulation that can treat wide variations. The main result is that, if there is no upper bound on the number of transportations (river crossings), a large class of subproblems can be solved in polynomialtime even when the passenger capacity of the boat is arbitrarily large. The authors speculated this fact at FUN 2012. On the other hand, this paper also demonstrates that, if an upper bound on the number of transportations is given, the problem is NP-hard even when the boat capacity is three, although a large class of subproblems can be solved in polynomialtime if the boat capacity is two.
We revisit the classical resource allocation problem with general convex objective functions, subject to an integer knapsack constraint. This class of problems is fundamental in discrete optimization and arises in a w...
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We revisit the classical resource allocation problem with general convex objective functions, subject to an integer knapsack constraint. This class of problems is fundamental in discrete optimization and arises in a wide variety of applications. In this paper, we propose a novel polynomial-time divide-and-conquer algorithm (called the multi-phase algorithm) and prove that it has a computational complexity of O(n log n log N), which outperforms the best known polynomial-time algorithm with O(n( log N)(2)). (C) 2015 Elsevier B.V. All rights reserved.
We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our ana...
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We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average-case hypothesis on the hardness of refuting satisfiability of random formulas.
We are given a set of items, and a set of knapsacks. Both the weight and the profit of an item are functions of the knapsack, and each knapsack has a positive real capacity. A restriction is setting that the number of...
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ISBN:
(纸本)9781479986460
We are given a set of items, and a set of knapsacks. Both the weight and the profit of an item are functions of the knapsack, and each knapsack has a positive real capacity. A restriction is setting that the number of the items which are admissible to each knapsack is no more than k, and these items are taken as input for each knapsack. We consider two following objectives: (1) maximizing the total profit of all the knapsacks (Max-Sum k-GMK);(2) maximizing the minimum profit of all the knapsacks (Max-Min k-GMK). We show that the two problems are NP-complete when k is greater than or equal t to 4. For the Max-Sum k-GMK problem, we can obtain a 1/2-approximation algorithm, and especially when k=2, we design an optimal algorithm. For the Max-Min k-GMK problem, we present a 1/(k-1)-approximation algorithm, and especially when k=2, this algorithm is an optimal algorithm.
We study multivariate linear problems in the average case setting with respect to a zero-mean Gaussian measure whose covariance kernel has a finite-order weights structure. This means that the measure is concentrated ...
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We study multivariate linear problems in the average case setting with respect to a zero-mean Gaussian measure whose covariance kernel has a finite-order weights structure. This means that the measure is concentrated on a Banach space of d-variate functions that are sums of functions of at most q* variables and the influence of each such term depends on a given weight. Here q* is fixed whereas d varies and can be arbitrarily large. For arbitrary finite-order weights, based on Smolyak's algorithm, we construct polynomial-time algorithms that use standard information. That is, algorithms that solve the d-variate problem to within e using of order epsilon(-p)d(q)* function values modulo a power of ln epsilon(-1). Here p is the exponent which measures the difficulty of the univariate (d = 1) problem, and the power of ln epsilon(-1) is independent of d. We also present a necessary and sufficient condition on finite-order weights for which we obtain strongly polynomial-time algorithms, i.e., when the number of function values is independent of d and polynomial in epsilon(-1). The exponent of epsilon(-1) may be, however, larger than p. We illustrate the results by two multivariate problems: integration and function approximation. For the univariate case we assume the r-folded Wiener measure. Then p = 1/(r + 1) for integration and p = 1/(r + 1/2) for approximation.
Given a multihop wireless network and a source-destination pair of nodes, this paper addresses the problem of jointly selecting a communication route and allocating transmit power levels, so that the end-to-end spectr...
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Given a multihop wireless network and a source-destination pair of nodes, this paper addresses the problem of jointly selecting a communication route and allocating transmit power levels, so that the end-to-end spectral efficiency of the route exceeds a desired threshold. Spectral-efficient routing has been subject to interest in the recent literature. The transmit power level, however, has been assumed to be known, and route selection was considered in isolation. This paper presents the first rigourously proven optimal, polynomial-time algorithms for two versions of the joint spectral-efficient routing and power allocation problem: sum-power minimization and maximum power minimization. The proposed algorithms rely on the divide-and-conquer principle and the Bellman-Ford algorithm for shortest (or widest) path computation. Our computational results further illustrate the efficiency of the proposed approach.
We consider the problem of scheduling unit-length jobs on identical machines subject to precedence constraints. We show that natural scheduling rules fail when the precedence constraints form a collection of stars or ...
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We consider the problem of scheduling unit-length jobs on identical machines subject to precedence constraints. We show that natural scheduling rules fail when the precedence constraints form a collection of stars or a collection of complete bipartite graphs. We prove that the problem is in fact NP-hard on collections of stars when the input is given in a compact encoding, whereas it can be solved in polynomialtime with standard adjacency list encoding. On a subclass of collections of stars and on collections of complete bipartite graphs we show that the problem can be solved in polynomialtime even when the input is given in compact encoding, in both cases via non-trivial algorithms. (C) 2014 Elsevier B.V. All rights reserved.
We propose a simple O(vertical bar n(5)/log n vertical bar L) algorithm for linear programming feasibility, that can be considered as a polynomial-time implementation of the relaxation method. Our work draws from Chub...
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We propose a simple O(vertical bar n(5)/log n vertical bar L) algorithm for linear programming feasibility, that can be considered as a polynomial-time implementation of the relaxation method. Our work draws from Chubanov's "Divide-and-Conquer" algorithm (Chubanov, 2012), with the recursion replaced by a simple and more efficient iterative method. A similar approach was used in a more recent paper of Chubanov (2013). (C) 2014 Elsevier B.V. All rights reserved.
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