Browsing by Subject "Polynomial algorithms"

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    Convexity and logical analysis of data
    (Elsevier, 2000) Ekin, O.; Hammer, P. L.; Kogan, A.
    A Boolean function is called k-convex if for any pair x,y of its true points at Hamming distance at most k, every point "between" x and y is also true. Given a set of true points and a set of false points, the central question of Logical Analysis of Data is the study of those Boolean functions whose values agree with those of the given points. In this paper we examine data sets which admit k-convex Boolean extensions. We provide polynomial algorithms for finding a k-convex extension, if any, and for finding the maximum k for which a k-convex extension exists. We study the problem of uniqueness, and provide a polynomial algorithm for checking whether all k-convex extensions agree in a point outside the given data set. We estimate the number of k-convex Boolean functions, and show that for small k this number is doubly exponential. On the other hand, we also show that for large k the class of k-convex Boolean functions is PAC-learnable. (C) 2000 Elsevier Science B.V. All rights reserved.
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    A genuinely polynomial primal simplex algorithm for the assignment problem
    (Elsevier, 1993) Akgül, M.
    We present a primal simplex algorithm that solves the assignment problem in 1 2n(n+3)-4 pivots. Starting with a problem of size 1, we sequentially solve problems of size 2,3,4,...,n. The algorithm utilizes degeneracy by working with strongly feasible trees and employs Dantzig's rule for entering edges for the subproblem. The number of nondegenerate simplex pivots is bounded by n-1. The number of consecutive degenerate simplex pivots is bounded by 1 2(n-2)(n+1). All three bounds are sharp. The algorithm can be implemented to run in O(n3) time for dense graphs. For sparse graphs, using state of the art data structures, it runs in O(n2 log n+nm) time, where the bipartite graph has 2n nodes and m edges. © 1993.
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    A sequential dual simplex algorithm for the linear assignment problem
    (Elsevier, 1988) Akgül, M.
    We present a sequential dual-simplex algorithm for the linear problem which has the same complexity as the algorithms of Balinski [3,4] and Goldfarb [8]: O(n2) pivots, O(n2 log n + nm) time. Our algorithm works with the (dual) strongly feasible trees and can handle rectangular systems quite naturally.