Norton, Plotkin and Tardos proved that-loosely spoken, an LP problem is solvable in time O(Tq(k+1)) if deleting k fixed columns or rows, we obtain a problem which can be solved by an algorithm that makes at most T ste...
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Norton, Plotkin and Tardos proved that-loosely spoken, an LP problem is solvable in time O(Tq(k+1)) if deleting k fixed columns or rows, we obtain a problem which can be solved by an algorithm that makes at most T steps and q comparisons. This paper improves this running time to O(Tq(k)). (C) 2004 Elsevier B.V. All rights reserved.
Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approa...
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In this paper, we first consider a network improvement problem, called vertex-to-vertices distance reduction problem. The problem is how to use a minimum cost to reduce lengths of the edges in a network so that the di...
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In this paper, we first consider a network improvement problem, called vertex-to-vertices distance reduction problem. The problem is how to use a minimum cost to reduce lengths of the edges in a network so that the distances from a given vertex to all other vertices are within a given upper bound. We use l(infinity), l(1) and l(2) norms to measure the total modification cost respectively. Under l(infinity) norm, we present a stronglypolynomial algorithm to solve the problem, and under l(1) or weighted l(2) norm, we show that achieving an approximation ratio O( log(|V|)) is NP-hard. We also extend the results to the vertex-to-points distance reduction problem, which is to reduce the lengths of edges most economically so that the distances from a given vertex to all points on the edges of the network are within a given upper bound.
Problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements, or order constraints. In such problems, the objective is to minimize the deviation cost func...
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Problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements, or order constraints. In such problems, the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modify value. In Markov random field problems, there is also a pairwise relationship between the objects. The objective in Markov random field problem is to minimize the sum of the deviation cost function and a penalty function that grows with the distance between the values of related pairs-separation function. We discuss Markov random fields problems in the context of a representative application-the image segmentation problem. In this problem, the goal is to modify color shades assigned to pixels of an image so that the penalty function consisting of one term due to the deviation from the initial color shade and a second term that penalizes differences in assigned values to neighboring pixels is minimized. We present here an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear;and in stronglypolynomial time when the deviation cost function is linear, quadratic or piecewise linear convex with few pieces (where "few" means a number exponential in a polynomial function of the number of variables and constraints). The complexity of the algorithm for a problem on n pixels or variables, m adjacency relations or constraints, and range of variable values (colors) U, is O(T (n, m) + n log U) where T (n, m) is the complexity of solving the minimum s, t cut problem on a graph with n nodes and m arcs. Furthermore, other algorithms are shown to solve the problem with convex deviation and convex separation in running time O (mn log n log n U) and the problem with nonconvex deviation and convex separation in running time 0(T (n U, m U). The nonconvex separation problem is NP-hard even for fixed value of U. For the family of p
In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficie...
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In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficient vector c of the objective function, optimally and within bounds, such that x* becomes an optimal solution of the linear program. It is shown that the problem can be formulated as a combinatorial linear program and can be transformed further into a minimum cost circulation problem. Hence, it can be solved in stronglypolynomial time. We also give a necessary and sufficient condition for the feasibility of the problem. Finally, we extend the discussion to the version of the inverse problem with multiple feasible solutions.
The cycle-canceling algorithm is one of the earliest algorithms to solve the minimum-cost flow problem. This algorithm maintains a feasible solution x in the network G and proceeds by augmenting flows along negative-c...
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The cycle-canceling algorithm is one of the earliest algorithms to solve the minimum-cost flow problem. This algorithm maintains a feasible solution x in the network G and proceeds by augmenting flows along negative-cost directed cycles in the residual network G(x) and thereby canceling them. For the minimum-cost flow problem with integral data, the generic version of the cycle-canceling algorithm runs in pseudopolynomial time, but several polynomial-time specific implementations can be obtained by specifying the choices of cycles to be canceled. In this paper, we describe a new polynomial-time implementation of the cycle-canceling algorithm. Our algorithm is a scaling algorithm and proceeds by augmenting flows along negative cycles with "sufficiently large" residual capacity. Further, it identifies such a cycle by solving a shortest path problem with nonnegative are lengths. For a network with n nodes and m arcs, our cycle-canceling algorithm performs O(m log(nU)) augmentations and runs in O(m(m + n log n) log (nU)) time, where U is an upper bound on the node supplies/demands and finite are capacities. We also show that the cycle-canceling algorithm (i) can solve the uncapacitated minimum-cost flow problem in O(n(m + n log n) log (nU)) time;(ii) can obtain an integer optimal solution of the convex cost-flow problem in O(m(m + n log n) log (nU)) time;and (iii) can be modified so that it runs in O(m(m + n log n) min (log (nU), m log n)) time, which is a stronglypolynomial time bound. (C) 2000 John Wiley & Sons, Inc.
We introduce the concept of a characterization set for the nucleolus of a cooperative game and develop sufficient conditions for a collection of coalitions to form a characterization set thereof. Further, we formalize...
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We introduce the concept of a characterization set for the nucleolus of a cooperative game and develop sufficient conditions for a collection of coalitions to form a characterization set thereof. Further, we formalize Kopelowitz's method for computing the nucleolus through the notion of a sequential LP process, and derive a general relationship between the size of a characterization set and the complexity of computing the nucleolus.
In this paper a method for solving perfect systems of linear inequalities is presented. It is based on selecting and removing inessential constraints. This method is a stronglypolynomial one for the class of systems ...
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We present two efficient algorithms for the minimum-cost flow problem in which are costs are piecewise-linear and convex. Our algorithms are based on novel algorithms of Orlin, which were developed for the case of lin...
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We present two efficient algorithms for the minimum-cost flow problem in which are costs are piecewise-linear and convex. Our algorithms are based on novel algorithms of Orlin, which were developed for the case of linear are costs. Our first algorithm uses the Edmonds-Karp scaling technique. Its complexity is O(M log U(m + n log M)) for a network with n vertices, m arcs, M linear cost segments, and an upper bound U on the supplies and the capacities. The second algorithm is a stronglypolynomial version of the first, and it uses Tardos's idea of contraction. Its complexity is O(M log M(m + n log M)). Both algorithms improve by a factor of at least Mim the complexity of directly applying existing algorithms to a transformed network in which are costs are linear.
We prove a tight Theta(min(nm log(nC), nm(2))) bound on the number of iterations of the minimum-mean cycle-canceling algorithm of Goldberg and Tarjan [13]. We do this by giving the lower bound and by improving the str...
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We prove a tight Theta(min(nm log(nC), nm(2))) bound on the number of iterations of the minimum-mean cycle-canceling algorithm of Goldberg and Tarjan [13]. We do this by giving the lower bound and by improving the stronglypolynomial upper bound on the number of iterations to O(nm(2)). We also give an improved version of the maximum-mean cut canceling algorithm of [7], which is a dual of the minimum-mean cycle-canceling algorithm. Our version of the dual algorithm runs in O(nm(2)) iterations.
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