On approximability of linear ordering and related NP-optimization problems on graphs

Mishra, Sambit Kumar; Sikdar, Kripasindhu

doi:10.1016/s0166-218x(03)00444-x

Cited by 8 publications

(3 citation statements)

References 27 publications

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“…We work with instances of the permutation flowshop scheduling problem (PFSP), the linear ordering problem (LOP), and the quadratic assignment problem (QAP), which are known to be NP-hard problems (Garey et al, 1976;Mishra and Sikdar, 2004;Sahni and Gonzalez, 1976). The flowshop scheduling problem can be stated as follows: there are b jobs to be scheduled in c machines.…”

Section: Experimental Designmentioning

confidence: 99%

Anatomy of the Attraction Basins: Breaking with the Intuition

Hernando

Mendiburu

Lozano

2019

Evolutionary Computation

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Solving combinatorial optimization problems efficiently requires the development of algorithms that consider the specific properties of the problems. In this sense, local search algorithms are designed over a neighborhood structure that partially accounts for these properties. Considering a neighborhood, the space is usually interpreted as a natural landscape, with valleys and mountains. Under this perception, it is commonly believed that, if maximizing, the solutions located in the slopes of the same mountain belong to the same attraction basin, with the peaks of the mountains being the local optima. Unfortunately, this is a widespread erroneous visualization of a combinatorial landscape. Thus, our aim is to clarify this aspect, providing a detailed analysis of, first, the existence of plateaus where the local optima are involved, and second, the properties that define the topology of the attraction basins, picturing a reliable visualization of the landscapes. Some of the features explored in this article have never been examined before. Hence, new findings about the structure of the attraction basins are shown. The study is focused on instances of permutation-based combinatorial optimization problems considering the 2-exchange and the insert neighborhoods. As a consequence of this work, we break away from the extended belief about the anatomy of attraction basins.

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Section: Experimental Designmentioning

confidence: 99%

Anatomy of the Attraction Basins: Breaking with the Intuition

Hernando

Mendiburu

Lozano

2019

Evolutionary Computation

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“…Unfortunately, the problem is NP-hard. Furthermore, it is known to be APX-complete, meaning that there is no polynomial time approximation scheme unless P=NP (Mishra and Sikdar, 2004). However, there are various heuristic procedures for approximating it (Tromble, 2009).…”

Section: The Linear Ordering Problemmentioning

confidence: 99%

Learning linear ordering problems for better translation

Tromble

Eisner

2009

Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing Volume 2 - EMNLP '09

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We apply machine learning to the Linear Ordering Problem in order to learn sentence-specific reordering models for machine translation. We demonstrate that even when these models are used as a mere preprocessing step for German-English translation, they significantly outperform Moses' integrated lexicalized reordering model. Our models are trained on automatically aligned bitext. Their form is simple but novel. They assess, based on features of the input sentence, how strongly each pair of input word tokens w i , w j would like to reverse their relative order. Combining all these pairwise preferences to find the best global reordering is NP-hard. However, we present a non-trivial O(n 3) algorithm, based on chart parsing, that at least finds the best reordering within a certain exponentially large neighborhood. We show how to iterate this reordering process within a local search algorithm, which we use in training.

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“…Given a matrix of numerical values, the linear ordering problem consists of finding a simultaneous permutation of the rows and columns, such that the sum of the entries above the main diagonal is maximized. This problem has attracted the attention of researchers because it has very special properties that make it apparently really easy to solve; however, it is an NP-hard problem (Mishra and Sikdar, 2004). For example, the symmetry enclosing this problem should be highlighted (Marti and Reinelt, 2011;Hernando et al, 2019b): the solution that maximizes the objective function is the reverse of the one that minimizes it, or in general, the reverse of the k-th best solution is the k-th worst solution.…”

Section: Introductionmentioning

confidence: 99%

Journey to the center of the linear ordering problem

Hernando

Mendiburu

Lozano

2020

Proceedings of the 2020 Genetic and Evolutionary Computation Conference

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A number of local search based algorithms have been designed to escape from the local optima, such as, iterated local search or variable neighborhood search. The neighborhood chosen for the local search as well as the escape technique play a key role in the performance of these algorithms. Of course, a specific strategy has a different effect on distinct problems or instances. In this paper, we focus on a permutation-based combinatorial optimization problem: the linear ordering problem. We provide a theoretical landscape analysis for the adjacent swap, the swap and the insert neighborhoods. By making connections to other different problems found in the Combinatorics field, we prove that there are some moves in the local optima that will necessarily return a worse or equal solution. The number of these non-better solutions that could be avoided by the escape techniques is considerably large with respect to the number of neighbors. This is a valuable information that can be included in any of those algorithms designed to escape from the local optima, increasing their efficiency.

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On approximability of linear ordering and related NP-optimization problems on graphs

Cited by 8 publications

References 27 publications

Anatomy of the Attraction Basins: Breaking with the Intuition

Anatomy of the Attraction Basins: Breaking with the Intuition

Learning linear ordering problems for better translation

Journey to the center of the linear ordering problem

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