Wyniki wyszukiwania - BazTech

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Polynomial-time Classification of Skew-symmetrizable Matrices with a Positive Definite Quasi-Cartan Companion

Pérez Claudia, Rivera Daniel

Fundamenta Informaticae

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2021

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Vol. 181, nr 4

313--337

EN

Skew-symmetrizable matrices play an essential role in the classification of cluster algebras. We prove that the problem of assigning a positive definite quasi-Cartan companion to a skew-symmetrizable matrix is in polynomial class P. We also present an algorithm to determine the finite type Δ ∈ { {𝔸n, 𝔻n, 𝔹n, ℂn, 𝔼6, 𝔼7, 𝔼8, 𝔽4, 𝔾2 } of a cluster algebra associated to the mutation-equivalence class of a connected skew-symmetrizable matrix B , if it has one.

2

Better polynomial algorithms for scheduling unit-length jobswith bipartite incompatibility graphs on uniform machines

Pikies T., Kubale Marek

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 1

31--36

EN

The goal of this paper is to explore and to provide tools for the investigation of the problems of unit-length scheduling of incompatible jobs on uniform machines. We present two new algorithms that are a significant improvement over the known algorithms. The first one is Algorithm 2 which is 2-approximate for the problem Qm|pj = 1, G = bisubquartic|Cmax. The second one is Algorithm 3 which is 4-approximate for the problem Qm|pj = 1, G = bisubquartic|ΣCj, where m ϵ {2, 3, 4}. The theory behind the proposed algorithms is based on the properties of 2-coloring with maximal coloring width, and on the properties of ideal machine, an abstract machine that we introduce in this paper.

3

Scheduling of Identical Jobs with Bipartite Incompatibility Graphs on Uniform Machines. Computational Experiments

Duraj S., Kopeć P., Kubale M., Pikies T.

Decision Making in Manufacturing and Services

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2017

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Vol. 11, no. 1-2

53--61

EN

In this paper, we consider the problem of scheduling unit-length jobs on three or four uniform parallel machines to minimize the schedule length or total completion time. We assume that the jobs are subject to some types of mutual exclusion constraints, modeled by a bipartite graph of a bounded degree. The edges of the graph correspond to the pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard, we show that our problem can be solved to optimality in polynomial time under some restrictions imposed on the number of machines, their speeds, and the structure of the incompatibility graph. The theoretical considerations are accompanied by computer experiments with a certain model of scheduling.

4

Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

Furmańczyk H., Kubale M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2017

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Vol. 65, nr 1

29--34

EN

In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s1 ≥ s2 ≥ s3 ≥ s4, respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s1 = s2 = s3. If, however, Δ ≤ 4 and s1 ≥ 12s2, s2 = s3 = s4, then the problem can be solved to optimality in time O(n1.5). The same algorithm returns a solution of value at most 2 times optimal provided that s1 ≥ 2s2. Finally, we study the case s1 ≥ s2 ≥ s3 = s4 and give a 32/15-approximation algorithm running also in O(n1.5) time.

5

Modele i metody kolorowania grafów. Część II

Kubale M.

Przegląd Elektrotechniczny

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2012

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R. 88, nr 11a

51-55

PL

Niniejszy artykuł jest drugą częścią 2-odcinkowego cyklu przeglądowego na temat modeli i metod kolorowania grafów. Przedstawiono w nim najważniejsze, z punktu widzenia zastosowań, modele kolorowania grafów. W szczególności pokazano różne kryteria i ograniczenia modyfikujące kolorowanie klasyczne. Ponieważ kolorowanie we wszystkich tych odmianach i wariantach jest NP-trudne, podano oszacowania na liczbę chromatyczną (indeks chromatyczny) oraz podano potencjalne zastosowania rozważanych modeli w problemach naukowo-technicznych.

EN

This is the second of a couple of review papers on models and methods of graph coloring. We present herein the main models of graph coloring from a practical point of view. In particular, we show various criteria and modifications of classical coloring model. Since graph coloring is NP-hard in various modifications and variants, we give simple bounds on the chromatic number (chromatic index) as well as we give potential applications of the chromatic methods in science and technology.

6

Szeregowanie rozrzedzonych systemów zadań jednostkowych 1- i 2-procesowych w oknach czasowych

Giaro K., Kubale M.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2005

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T. 9, z. 1-2

85-94

PL

W artykule rozważono rozrzedzone systemy niepodzielnych zadań 1- i 2-procesorowych o jednostkowych czasach wykonywania. Przedstawiono wielomianowe algorytmy wykorzystujące programowanie dynamiczne, pozwalające na znalezienie optymalnego uszeregowania względem szerokiej rodziny funkcji kryterialnych. Stopień rozrzedzenia systemu zdefiniowano, posługując się jego modelem grafowym - w zakresie naszego zainteresowania leżą jedynie takie instancje problemów szeregowania, których modelami są grafy o ograniczonej liczbie cyklomatycznej. Istotnym elementem opracowanych procedur są algorytmy rozwiązujące pewne zagadnienia związane z wyszukiwaniem skojarzeń w grafach.

EN

In the paper sparse systems of dedicated 1- and 2-processor tasks with unit execution times are considered. Polynomial-time algorithms based on dynamic programming are given. These algorithms allow finding optimal solutions with respect to broad range of criterion functions. The sparsity of a system is measured in terms of the number of edges in the corresponding scheduling graph. More precisely, we are focused on graphs whose cyclomatic number is bounded by a constant. Our algorithms invoke procedures for finding maximal matching in graphs.