The main difficulty in solving the systems of linear Diophantine equations is the very rapid growth of the intermediate results. Hence, it is important to design algorithms that restraint the growth of intermediate re...
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The main difficulty in solving the systems of linear Diophantine equations is the very rapid growth of the intermediate results. Hence, it is important to design algorithms that restraint the growth of intermediate results. The use of the basis reduction algorithm for solving a linear Diophantine equation is examined in this paper. The motivation behind choosing the basis reduction is twofold. First the basis reduction allows us to work only with integers, which avoids the round-off problems. Second, the basis reduction finds short and nearly orthogonal vectors belonging to the lattice described by the basis. Then, we expect the general solution obtained by this basis will also be short. It is important to note that the basis reduction does not change the lattice, it only derives an alternative way of spanning it. Once we have obtained the vectors given by the reduced basis, we use them to find the general solution of the linear Diophantine equation.
In this paper we consider several single-machine scheduling problems with general learning effects. By general learning effects, we mean that the processing time of a job depends not only on its scheduled position, bu...
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In this paper we consider several single-machine scheduling problems with general learning effects. By general learning effects, we mean that the processing time of a job depends not only on its scheduled position, but also on the total normal processing time of the jobs already processed. We show that the scheduling problems of minimization of the make-span, the total completion time and the sum of the 0th (0 >= 0) power of job completion times can be solved in polynomialtime under the proposed models. We also prove that some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomialtime. (C) 2012 Elsevier Inc. All rights reserved.
For a matrix W \in Zmx n, m \leq n, and a convex function g : Rm \rightarrowR, we are interested in minimizing f(x) = g(Wx) over the set {0, 1\} n. We will study separable convex functions and sharp convex functions g...
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For a matrix W \in Zmx n, m \leq n, and a convex function g : Rm \rightarrowR, we are interested in minimizing f(x) = g(Wx) over the set {0, 1\} n. We will study separable convex functions and sharp convex functions g. Moreover, the matrix W is unknown to us. Only the number of rows m \leq n and II W II,, are revealed. The composite function f (x) is presented by a zeroth and first order oracle only. Our main result is a proximity theorem that ensures that an integral minimum and a continuous minimum for separable convex and sharp convex functions are always ``close"" by. This will be a key ingredient in developing an algorithm for detecting an integer minimum that achieves a running time of roughly (mII W II,,)O(m3) \cdot poly(n). In the special case when (i) W is given explicitly and (ii) g is separable convex one can also adapt an algorithm of Hochbaum and Shanthikumar [J. ACM, 37 (1990), pp. 843--862]. The running time of this adapted algorithm matches the running time of our general algorithm.
Let G = (V, E) be a graph. A subset D of vertices of G is called a dominating set of G if for every u epsilon V \ D, there exists a vertex v epsilon D such that uv epsilon E. A dominating set D of a graph G is called ...
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Let G = (V, E) be a graph. A subset D of vertices of G is called a dominating set of G if for every u epsilon V \ D, there exists a vertex v epsilon D such that uv epsilon E. A dominating set D of a graph G is called a secure dominating set of G if for every u epsilon V \ D, there exists a vertex V epsilon D such that uv epsilon E and (D \ {v}) boolean OR {u} is a dominating set of G. The secure domination number of G, denoted by gamma(s)(G), is the minimum cardinality of a secure dominating set of G. In this paper, we present an 0 (n + m) timealgorithm to compute the secure domination number of a cograph having n vertices and m edges. (C) 2019 Elsevier B.V. All rights reserved.
A polynomial time algorithm for computing the automorphism group of cyclic automata is given. The problems of computing the automorphism group of automata, testing either two graphs, two strongly connected automata fo...
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A polynomial time algorithm for computing the automorphism group of cyclic automata is given. The problems of computing the automorphism group of automata, testing either two graphs, two strongly connected automata for isomorphism, or two automata for strong-isomorphism are shown to be polynomially equivalent.
extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption...
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extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed;some of these are fast to compute, while others are known to achieve the mini-max optimal rate over certain Gaussian or sub-Gaussian classes. In this paper, we show that, under a widely-believed assumption from computational complexity theory, there is a fundamental trade-off between statistical and computational performance in this problem. More precisely, working with new, larger classes satisfying a restricted covariance concentration condition, we show that there is an effective sample size regime in which no randomised polynomial time algorithm can achieve the minimax optimal rate. We also study the theoretical performance of a (polynomialtime) variant of the well-known semidefinite relaxation estimator, revealing a subtle interplay between statistical and computational efficiency.
Let G be a finite, nilpotent group. The computational complexity of the equation solvability problem over G is known to be in P. The complexity is understood in the length of the equation over G. So far the fastest al...
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Let G be a finite, nilpotent group. The computational complexity of the equation solvability problem over G is known to be in P. The complexity is understood in the length of the equation over G. So far the fastest algorithm to decide whether or not an equation of length n has a solution over G was running in O (n(|G||G|...|G||G|)) time. Here the height of the tower is the nilpotency class of G. We prove that one can decide in O (n(1/2) (|G|2 log|G|)) time whether an equation of length n has a solution over G. The key ingredient of the proof is to represent group expressions using the polycyclic presentation of p-groups. (C) 2017 Elsevier Inc. All rights reserved.
The paper is concerned with the best approximation of a piecewise constant univariate function with n constant pieces by a similar function with at most k constant pieces, where k < n. We show that for the case of ...
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The paper is concerned with the best approximation of a piecewise constant univariate function with n constant pieces by a similar function with at most k constant pieces, where k < n. We show that for the case of the maximum norm, the general dynamic programming algorithm of complexity O(kn2) can be considerably improved.
We consider a hierarchical optimization problem for imprecise computation tasks, where each task is weighted with two weights, w and w'. The primary criterion is to minimize the total w-weighted error of all optio...
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We consider a hierarchical optimization problem for imprecise computation tasks, where each task is weighted with two weights, w and w'. The primary criterion is to minimize the total w-weighted error of all optional parts of tasks and the secondary criterion is to minimize the maximum w-weighted error. An algorithm is given with time complexity O(kn(3) log(2)n) for parallel and identical processors and O(kn(2)) for a single processor, where k is the number of distinct w-weights.
We investigate a single-machine scheduling problem,where a deteriorating rate-modifying activity can be performed on the machine to reduce the processing times of *** objective is to minimize the number of tardy *** t...
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We investigate a single-machine scheduling problem,where a deteriorating rate-modifying activity can be performed on the machine to reduce the processing times of *** objective is to minimize the number of tardy *** the assumption that the duration of the rate-modifying activity is a nonnegative and nondecreasing function on its starting time,we propose an optimal polynomial time algorithm running in O(n^(3))time.
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