Packing Squares into Squares

We consider the problem of packing a list of squares into unit capacity squares. The objective of this problem is to minimize the number of unit capacity squares used in the packing. We consider a restricted version of this problem and exhibit a polynomial time algorithm to solve it. We use this algorithm in the design of an approximation algorithm for the original problem, and show that its asymptotic performance bound is 1:988. This bound compares favourably with the bound 2:125, known for an algorithm obtained by Chung, Garey and Johnson for the more general problem, where the items to be packed are rectangles. 1 Introduction We consider the problem of packing squares into squares, dened as follows. Given a list L of n squares and (an unlimited quantity of) squares Q, called bins, pack the n squares of L into a minimum number of bins. We note that there is no loss of generality to assume that the bins have unit capacity (width and height 1), since otherwise we can scale the squar..

Prices of anarchy of selfish 2D bin packing games

We consider a game-theoretical problem called selfish 2-dimensional bin packing game, a generalization of the 1-dimensional case already treated in the literature. In this game, the items to be packed are rectangles, and the bins are unit squares. The game starts with a set of items arbitrarily packed in bins. The cost of an item is defined as the ratio between its area and the total occupied area of the respective bin. Each item is a selfish player that wants to minimize its cost. A migration of an item to another bin is allowed only when its cost is decreased. We show that this game always converges to a Nash equilibrium (a stable packing where no single item can decrease its cost by migrating to another bin). We show that the pure price of anarchy of this game is unbounded, so we address the particular case where all items are squares. We show that the pure price of anarchy of the selfish square packing game is at least 2.3634 and at most 2.6875. We also present analogous results for the strong Nash equilibrium (a stable packing where no nonempty set of items can simultaneously migrate to another common bin and decrease the cost of each item in the set). We show that the strong price of anarchy when all items are squares is at least 2.0747 and at most 2.36053003355374CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP456792/2014-7; 311499/2014-7; 305705/2014-8; 308116/2016-0; 306464/2016-0; 425340/2016-32013/03447-6; 2015/11937-9; 2016/01860-1; 2016/23552-

Clustering through continuous facility location problems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We consider the Continuous Facility Location Problem (ConFLP). Given a finite set of clients C subset of R-d and a number f is an element of R+, ConFLP consists in opening a set F' subset of R-d of facilities, each at cost f, and connecting each client to an open facility. The objective is to minimize the costs of opening facilities and connecting clients. We reduce ConFLP to the standard Facility Location Problem (FLP), by using the so-called approximate center sets. This reduction preserves the approximation, except for an error epsilon, and implies 1.488 + epsilon and 2.04 + epsilon-approximations when the connection cost is given by the Euclidean distance and the squared Euclidean distance, respectively. Moreover, we obtain approximate center sets for the case that the connection cost is the ath power of the Euclidean distance, achieving approximations for the corresponding problems, for any alpha >= 1. As a byproduct, we also obtain a polynomial-time approximation scheme for the k-median problem with this cost function, for any fixed k. (C) 2016 Elsevier B.V. All rights reserved.We consider the Continuous Facility Location Problem (ConFLP). Given a finite set of clients C subset of R-d and a number f is an element of R+, ConFLP consists in opening a set F' subset of R-d of facilities, each at cost f, and connecting each client to657B137145FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)2010/20710-4477692/2012-5; 311499/2014-

An extension of Queiroz and Miyazawa's method for vertical stability in two-dimensional packing problems to deal with horizontal stability

A method to handle the cargo horizontal stability in two-dimensional packing problems is proposed. Mechanical equilibrium concepts are used to assess the cargo stability at which vertical and horizontal forces act on packing. The proposed method improves the methods based on either a support factor for an item's lateral sides or the number of supporting sides that cannot guarantee the stability. The method deals with the horizontal stability for which there is no other method based on the mechanical equilibrium. It is proved that the proposed method has the worst-case time complexity of , therefore improving a previous result in the literature. Numerical experiments are provided over instances of the two-dimensional knapsack problem. For that, an exact two-level algorithm is developed and it obtained the optimal stable solution of of the instances51610491070CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE GOIÁS - FAPEGFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP311499/2014-7; 308312/2016-3; 425340/2016-3sem informação2013/13815-0; 2015/11937-9; 2016/01860-1; 2016/23552-

Algorithms for 3D guillotine cutting problems: Unbounded knapsack, cutting stock and strip packing

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: unbounded knapsack, cutting stock and strip packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the unbounded 3D knapsack problem, we extend the recurrence formula proposed by [1] for the rectangular knapsack problem and present a dynamic programming algorithm that uses reduced raster points. We also consider a variant of the unbounded knapsack problem in which the cuts must be staged. For the 3D cutting stock problem and its variants in which the bins have different sizes (and the cuts must be staged), we present column generation-based algorithms. Modified versions of the algorithms for the 3D cutting stock problems with stages are then used to build algorithms for the 3D strip packing problem and its variants. The computational tests performed with the algorithms described in this paper indicate that they are useful to solve instances of moderate size. (C) 2011 Elsevier Ltd. All rights reserved.392200212Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

A systematic approach to bound factor-revealing LPs and its application to the metric and squared metric facility location problems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and show that each such program can be solved by a computer to bound the approximation factor of an associated algorithm. Obtaining an UPFRP is straightforward, and can be used as an alternative to analytical proofs, that are usually very long and tedious. We apply this technique to the metric facility location problem (MFLP) and to a generalization where the distance function is a squared metric. We call this generalization the squared metric facility location problem (SMFLP), and prove that there is no approximation factor better than 2.04, assuming P not equal NP. Then, we analyze the best known algorithms for the MFLP based on primal-dual and LP-rounding techniques when they are applied to the SMFLP. We prove very tight bounds for these algorithms, and show that the LP-rounding algorithm achieves a ratio of 2.04, and therefore has the best possible factor for the SMFLP. We use UPFRPs in the dualfitting analysis of the primal-dual algorithms for both the SMFLP and the MFLP, improving some of the previous analysis for the MFLP.A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and show that each such program can be solved by a computer to bound the approximation factor o1532655685CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULONUMEC - NÚCLEO DE APOIO À PESQUISA EM MODELAGEM ESTOCÁSTICA E COMPLEXIDADEConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [309657/2009-1, 306860/2010-4, 473867/2010-9, 477692/2012-5]FAPESP [2010/20710-4]309657/2009-1; 306860/2010-4; 473867/2010-9; 477692/2012-52010/20710-4sem informaçã

Polynomial-time approximation schemes for circle and other packing problems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)We consider the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height , for some arbitrarily small number . For this problem, we obtain an asymptotic approximation scheme (APTAS) that is polynomial on , and thus may be given as part of the problem input. For the special case that is constant, we give a (one dimensional) resource augmentation scheme, that is, we obtain a packing into bins of unit width and height using no more than the number of bins in an optimal packing without resource augmentation. Additionally, we obtain an APTAS for the circle strip packing problem, whose goal is to pack a set of circles into a strip of unit width and minimum height. Our algorithms are the first approximation schemes for circle packing problems, and are based on novel ideas of iteratively separating small and large items, and may be extended to a wide range of packing problems that satisfy certain conditions. These extensions comprise problems with different kinds of items, such as regular polygons, or with bins of different shapes, such as circles and spheres. As an example, we obtain APTAS's for the problems of packing d-dimensional spheres into hypercubes under the L-p-norm.We consider the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height , for some arbitrarily small number . For this problem, we obtain an asymptotic approximation sche762536568CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULONUMEC - NÚCLEO DE APOIO À PESQUISA EM MODELAGEM ESTOCÁSTICA E COMPLEXIDADEConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)303987/2010-3; 306860/2010-4; 477203/2012-4; 477692/2012-52010/20710-4; 2013/02434-8; 2013/03447-6; 2013/21744-8sem informaçã

Smart energy pricing for demand-side management in renewable energy smart grids

Smart grids are expected to provide various benefits to society by integrating advances in power engineering with recent developments in the field of information and communications technology. One of the advantages is the support to efficient demand-side management (DSM), for example, changes in consumer demands for energy based on using incentives. Indeed, DSM is expected to help grid operators balance time-varying generation by wind and solar units, and the optimization of their usage. This paper focuses on DSM considering renewable energy generation and proposes an auction, in which consumers submit bids to renewable energy usage plans. An additional model is introduced to allow consumers to compute their bid for a given usage plan. Both models have been extended to include energy storage devices. The proposed model is compared to a system with time-varying pricing for energy, where it is shown to allow consumers to use more appliances, to lead to a larger profit, and to reduce the peak-to-average ratio of energy consumption. Finally, the impact of the use of energy storage in households and in the energy provider is also consideredCONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQCOORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPESFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP311499/2014-7; 425340/2016-3; 477692/20125; 308689/2017-8não tem2013/21744-8; 2015/11937-9; 2016/01860-1; 2016/23552-

Strategies for protein folding and design

Fundamental challenges in molecular biology can be addressed by using simple models on a lattice, where statistical mechanics and combinatoric techniques can be employed. The basic premise is that it is sensible to test any proposed method on the simplest of models in order to assess their validity before launching a full-scale attack on realistic problems. In this paper we follow this strategy and we present different efficient schemes to perform protein design and to extract effective amino acid interaction potentials