Approximation algorithms for time constrained scheduling

Jansen, Klaus; Öhring, Sabine R.

doi:10.1007/3-540-60321-2_12

Cited by 8 publications

(8 citation statements)

References 10 publications

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“…For UWCC, with an argument similar to the analysis of Alg2(NF) and Alg2(FF) by Jansen and Ohring [24], one can show that COPY + NF and COPY + FF have the approximation factors of α + 2 and α + 1.7, respectively, where α is the approximation factor of the underlying uncapacitated solution. Epstein and Levin [13] and Berman et al [7] proved that COPY + FFD is an (α + 1.69)-approximation algorithm.…”

Section: From Uncapacitated To Capacitatedmentioning

confidence: 92%

“…When the points are uniformly weighted, Epstein et al [14] designed a 2-competitive online algorithm for this problem and proved that this is the best possible. Our problem in this case (with general weights) is a special form of the bin-packingwith-conflicts problem [13,24]. In this problem, a conflict graph among items is also given and we should not pack two conflicting items into a bin.…”

Section: Related Workmentioning

confidence: 99%

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On capacitated covering with unit balls

Ghasemi

Razzazi

2014

International Journal of Computer Mathematics

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In this paper, we consider the problem of capacitated covering with unit balls. In this problem, a set of weighted points in a metric space ( d , ρ) is given, and we want to cover them with a minimum number of the unit balls of that metric space provided that the total weight assigned to each unit ball is at most one. The problem is NP-hard as it generalizes the covering-with-unit-balls problem. We consider the problem in two cases: (1) the weight of each point can be split among several unit balls and (2) the unsplittable case. In the latter case, the problem is a generalization of the bin-packing problem even when d = 1, and thus it is not approximable under 1.5, unless P = NP. We design a polynomial-time approximation scheme (PTAS) for the splittable case when d is a fixed constant and ρ is an p metric. This also results in a PTAS for the unsplittable case when all the points have the same weight. We also analyse several natural algorithms for this problem and prove that they achieve constant approximation factors.

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Section: From Uncapacitated To Capacitatedmentioning

confidence: 92%

Section: Related Workmentioning

confidence: 99%

On capacitated covering with unit balls

Ghasemi

Razzazi

2014

International Journal of Computer Mathematics

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“…There are papers also about special conflict graphs, for example McCloskey and Shankar published results for the case of clique-graphs [19], Jansen andÖhring [14], and also Epstein and Levin [12] studied perfect conflict graphs, and bipartite graphs. Khanafer et al [15] investigated the two-dimensional variant of this problem.…”

Section: A Bódismentioning

confidence: 99%

“…The Bin Packing Problem with Conflicts (BPPC) model, investigated by several authors, like Jansen andÖhring [14], Jansen [13], Sadykov and Vanderbeck [20] etc., is very similar to the Generalized LIB constraint in the sense that both models use a conflict graph G = (V, E), where E is the set of edges so, that if (i, j) ∈ E, then the items i and j cannot be packed into the same bin.…”

Section: A Bódismentioning

confidence: 99%

Bin packing with directed stackability conflicts

Bódis

2015

Acta Universitatis Sapientiae, Informatica

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The Bin Packing problem is a well-known and highly investigated problem in the computer science: we have n items given with their sizes, and we want to assign them to unit capacity bins such, that we use the minimum number of bins.In this paper, some generalizations of this problem are considered, where there are some additional stackability constraints defining that certain items can or cannot be packed on each other. The corresponding model in the literature is the Bin Packing Problem with Conflicts (BPPC), where this additional constraint is defined by an undirected conflict graph having edges between items that cannot be assigned to the same bin. However, we show some practical cases, where this conflict is directed, meaning that the items can be assigned to the same bin, but only in a certain order. Two new models are introduced for this problem: Bin Packing Problem with Hanoi Conflicts (BPPHC) and Bin Packing Problem with Directed Conflicts (BPPDC). In this work, the connection of the three conflict models is examined in detail.We have investigated the complexity of the new models, mainly the BPPHC model, in the special case where each item have the same size. We also considered two cases depending on whether re-ordering the items is allowed or not.We show that for the online version of the BPPHC model with unit size items, every Any-Fit algorithm gives not better than 3 when it is forbidden for the optimum to re-order the items, even if only 2 stackability classes, called Hanoi classes, are applied. This lower bound is generalized for arbitrary number of Hanoi classes. However, we also prove, that asymptotically the First-Fit algorithm is 1-competitive for this case.Finally, we introduce an algorithm for the offline version of the BP-PHC model with unit size items, which has polynomial time complexity, if the number of the Hanoi classes and the capacity of the bins are constant.

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“…Outro problema que inclui o 2KPCé o problema de empacotamento em bins com conflitos proposto em [13] e investigado em [14], [15], [16]. A partir disto, notamos que o 2KPCé um problema de interesse prático, com poucas referências na literatura e que merece atenção por parte dos pesquisadores.…”

Section: A Trabalhos Correlatosunclassified

Approaches for the 2D 0-1 knapsack problem with conflict graphs

Queiroz¹,

Miyazawa

2013

2013 XXXIX Latin American Computing Conference (CLEI)

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This work deals with the 0-1 knapsack problem in its two-dimensional variant, when there is a conflict graph related to pairs of conflicting items. Conflicting items must not be packed together in a same bin. This problem also arises as a subproblem in the bin packing problem and in supply chain scenarious. We propose a heuristic that generates iteratively a solution using a so called greedy randomized procedure. In order to avoid local optima solutions, a penalization memory list is used, and several packing strategies under a two-dimensional grid of points are considered. The heuristic solutions are compared with those ones computed by means of an integer programming model, also proposed in this work and solved with CPLEX solver. The heuristic got optimal solutions for 75% of the instances in a lower CPU time compared with that to solve the integer model.Keywords-Two-dimensional 0-1 knapsack problem, conflict graph, heuristic, integer programming. I. INTRODUÇÃOO problema da mochila 0-1é um clássico problema de otimização combinatória e aparece em diversos contextos práticos, bem como um subproblema de outros problemas de otimização, por exemplo, no problema de empacotamento em bins [1]. Variantes deste problema também têm sido apresentadas na literatura e aparecem nos meios práticos, como no caso bidimensional (2D): empacotamento de paletes e corte de chapas [2]; e no caso tridimensional (3D): empacotamento em contêineres [3]. Vale ressaltar que o problema da mochila 0-1 e suas variantes em outras dimensões são NP-difíceis [4].Uma restrição no problema da mochila 0-1 envolve itens conflitantes, ou seja, itens que não podem ser empacotados em um mesmo recipiente (por exemplo, por questões tecnológicas como alimentos e produtos tóxicos) [5], [6]. Neste caso, devese escolher qual dos itens empacotar no recipiente, gerando um empacotamento sem itens conflitantes.Este trabalho investiga o problema da mochila 0-1 na sua variante 2D considerando um grafo de conflitos associado aos itens. Este grafo mantém arestas entre itens conflitantes de forma que no máximo um deles esteja empacotado no recipiente. Os itens são elementos retangulares com largura e comprimento definidos, bem como um valor/lucro associado. O recipiente tambémé retangular e tem dimensões previamente definidas. O objetivoé empacotar em umúnico recipiente um subconjunto não-conflitante de itens que maximize o lucro total. O empacotamento final deve obedecer as dimensões do recipiente e os itens devem ser empacotados de forma ortogonal aos lados do recipiente, não podem se sobrepor e nem serem rotacionados. Denominaremos este problema por 2KPC. A. Trabalhos CorrelatosA versão unidimensional do problema da mochila 0-1 com grafo de conflitos foi inicialmente estudada por Yamada et al.[5], [6]. Tais autores apresentaram estratégias gulosas e algoritmos exatos para o problema. A estratégia principal gera uma solução inicial por meio de uma heurística gulosa, sendo melhorada por um processo de busca em vizinhança. As soluções são usadas como limitantes para o...

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Approximation algorithms for time constrained scheduling

Cited by 8 publications

References 10 publications

On capacitated covering with unit balls

On capacitated covering with unit balls

Bin packing with directed stackability conflicts

Approaches for the 2D 0-1 knapsack problem with conflict graphs

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