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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Machine learning 18 (1995), S. 187-230 
    ISSN: 0885-6125
    Keywords: computational learning theory ; on-line learning ; mistake-bounded learning ; function learning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger range such as $$\mathbb{N}$$ or $$\mathbb{R}$$ . In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binary branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not involve any “learning” model). Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient learning algorithm for learning convex piecewise linear functions from $$\mathbb{R}^d $$ into $$\mathbb{R}$$ . Previously, the class of linear functions from $$\mathbb{R}^d $$ into $$\mathbb{R}$$ was the only class of functions with multidimensional domain that was known to be learnable within the rigorous framework of a formal model for online learning. Finally we give a sufficient condition for an arbitrary class $$\mathcal{F}$$ of functions from $$\mathbb{R}$$ into $$\mathbb{R}$$ that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from $$\mathcal{F}$$ . This allows us to exhibit a number of further nontrivial classes of functions from $$\mathbb{R}$$ into $$\mathbb{R}$$ for which there exist efficient learning algorithms.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Machine learning 18 (1995), S. 187-230 
    ISSN: 0885-6125
    Keywords: computational learning theory ; on-line learning ; mistake-bounded learning ; function learning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger range such as ℕ or ℝ. In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binary branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not involve any “learning” model). Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient learning algorithm for learning convex piecewise linear functions from ℝ d into ℝ. Previously, the class of linear functions from ℝ d into ℝ was the only class of functions with multidimensional domain that was known to be learnable within the rigorous framework of a formal model for online learning. Finally we give a sufficient condition for an arbitrary class $$\mathcal{F}$$ of functions from ℝ into ℝ that allows us to learn the class of all functions that can be written as the pointwise maximum ofk functions from $$\mathcal{F}$$ . This allows us to exhibit a number of further nontrivial classes of functions from ℝ into ℝ for which there exist efficient learning algorithms.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1436-4646
    Keywords: Quadratic assignment ; Special cases ; Polynomially solvable ; Anti-Monge matrices ; Toeplitz matrices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge—Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti-Monge—Toeplitz QAP: (Pl) The “Turbine Problem”, i.e. the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized. (P2) The Traveling Salesman Problem on symmetric Monge distance matrices. (P3) The arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time. We identify conditions on the Toeplitz matrixB that lead to a simple solution for the Anti-Monge—Toeplitz QAP: The optimal permutation can be given in advance without regarding the numerical values of the data. The resulting theorems generalize and unify several known results on problems (P1), (P2), and (P3). We also show that the Turbine Problem is NP-hard and consequently, that the Anti-Monge—Toeplitz QAP is NP-hard in general. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 82 (1998), S. 235-254 
    ISSN: 1436-4646
    Keywords: Single machine scheduling ; Traveling Salesman Problem ; Polynomial time algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper a one-machine scheduling model is analyzed wheren different jobs are classified intoK groups depending on which additional resource they require. The change-over time from one job to another consists of the removal time or of the set-up time of the two jobs. It is sequence-dependent in the sense that the change-over time is determined by whether or not the two jobs belong to the same group. The objective is to minimize the makespan. This problem can be modeled as an asymmetric Traveling Salesman Problem (TSP) with a specially structured distance matrix. For this problem we give a polynomial time solution algorithm that runs in O(n logn) time. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 82 (1998), S. 191-198 
    ISSN: 1436-4646
    Keywords: Scheduling ; Approximation algorithm ; Approximation scheme ; Worst-case analysis ; Open shop ; Makespan
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, we demonstrate the existence of a polynomial time approximation scheme for makespan minimization in the open shop scheduling problem with an arbitrary fixed numberm of machines. For the variant of the problem where the number of machines is part of the input, it is known that the existence of an approximation scheme would implyP = NP. Hence, our result draws a precise separating line between approximable cases (i.e., withm fixed) and non-approximable cases (i.e., withm part of the input) of this shop problem. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1436-4646
    Keywords: Key words: neighborhood – local search – search problem – Travelling Salesman Problem – Quadratic Assignment Problem – polynomial time algorithm – NP-completeness – combinatorial optimization ; Mathematics Subject Classification (1991): 90C27, 90C35, 68Q25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract. This paper deals with exponential neighborhoods for combinatorial optimization problems. Exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can be solved in polynomial time, and we investigate combinatorial and algorithmical questions related to such neighborhoods.¶First, we perform a careful study of exponential neighborhoods for the TSP. We investigate neighborhoods that can be defined in a simple way via assignments, matchings in bipartite graphs, partial orders, trees and other combinatorial structures. We identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms, and we also provide variants that slightly violate these properties and lead to NP-complete optimization problems. Whereas it is relatively easy to find exponential neighborhoods over which the TSP can be solved in polynomial time, the corresponding situation for the QAP looks pretty hopeless: Every exponential neighborhood that is considered in this paper provably leads to an NP-complete optimization problem for the QAP.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    BIT 33 (1993), S. 580-595 
    ISSN: 1572-9125
    Keywords: 05C05 ; 68Q25 ; 68R10 ; algorithmic complexity ; planar layouts ; geometry
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The computational complexity of the following type of problem is studied. Given a geometric graphG=(P, S) whereP is a set of points in the Euclidean plane andS a set of straight (closed) line segments between pairs of points inP, we want to know whetherG possesses a crossingfree subgraph of a special type. We analyze the problem of detecting crossingfree spanning trees, one factors and two factors in the plane. We also consider special restrictions on the slopes and on the lengths of the edges in the subgraphs.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Journal of combinatorial optimization 1 (1998), S. 355-365 
    ISSN: 1573-2886
    Keywords: scheduling ; open shop ; makespan ; on-line algorithm ; heuristic algorithm ; worst-case performance ratio ; worst-case guarantee
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate the problem of on-line scheduling two-machine open shops with the objective of minimizing the makespan.Jobs arrive independently over time, and the existence of a job is not known until its arrival. In the clairvoyant on-line model, the processing requirement of every job becomes fully known at the arrival of the job, while inthe non-clairvoyant on-line model, this processing requirement is notknown until the job is processed and completed.In both models, scheduling of a job is irrevocable. We study the two-machine open shop problem for both models in the preemptive and in the non-preemptive version. For each of the four variants, we provide an algorithm that is best possible with respect to the worst-case performance. In the clairvoyant on-line model, the best worst-case performance ratios are 5/4 (preemptive) and 3/2 (non-preemptive), and in the non-clairvoyant on-line model, they are 3/2 (preemptive and non-preemptive).
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Journal of combinatorial optimization 2 (1998), S. 333-350 
    ISSN: 1573-2886
    Keywords: travelling salesman problem ; subtour patching ; combinatorial optimization ; computational complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider traveling salesman problems (TSPs) with a permuted Monge matrix as cost matrix where the associated patching graph has a specially simple structure: a multistar, a multitree or a planar graph. In the case of multistars, we give a complete, concise and simplified presentation of Gaikov's theory. These results are then used for designing an O(m3 + mn) algorithm in the case of multitrees, where n is the number of cities and m is the number of subtours in an optimal assignment. Moreover we show that for planar patching graphs, the problem of finding an optimal subtour patching remains NP-complete.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Journal of combinatorial optimization 3 (1999), S. 51-58 
    ISSN: 1573-2886
    Keywords: Steiner tree ; Kalmanson matrix ; circulant matrix ; computational complexity ; graph algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate the computational complexity of two special cases of the Steiner tree problem where the distance matrix is a Kalmanson matrix or a circulant matrix, respectively. For Kalmanson matrices we develop an efficient polynomial time algorithm that is based on dynamic programming. For circulant matrices we give an $$\mathcal{N}\mathcal{P}$$ -hardness proof and thus establish computational intractability.
    Type of Medium: Electronic Resource
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