21 research outputs found

    A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game

    Full text link
    We prove a tight lower bound on the asymptotic performance ratio ρ\rho of the bounded space online dd-hypercube bin packing problem, solving an open question raised in 2005. In the classic dd-hypercube bin packing problem, we are given a sequence of dd-dimensional hypercubes and we have an unlimited number of bins, each of which is a dd-dimensional unit hypercube. The goal is to pack (orthogonally) the given hypercubes into the minimum possible number of bins, in such a way that no two hypercubes in the same bin overlap. The bounded space online dd-hypercube bin packing problem is a variant of the dd-hypercube bin packing problem, in which the hypercubes arrive online and each one must be packed in an open bin without the knowledge of the next hypercubes. Moreover, at each moment, only a constant number of open bins are allowed (whenever a new bin is used, it is considered open, and it remains so until it is considered closed, in which case, it is not allowed to accept new hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448] showed that ρ\rho is Ω(logd)\Omega(\log d) and O(d/logd)O(d/\log d), and conjectured that it is Θ(logd)\Theta(\log d). We show that ρ\rho is in fact Θ(d/logd)\Theta(d/\log d). To obtain this result, we elaborate on some ideas presented by those authors, and go one step further showing how to obtain better (offline) packings of certain special instances for which one knows how many bins any bounded space algorithm has to use. Our main contribution establishes the existence of such packings, for large enough dd, using probabilistic arguments. Such packings also lead to lower bounds for the prices of anarchy of the selfish dd-hypercube bin packing game. We present a lower bound of Ω(d/logd)\Omega(d/\log d) for the pure price of anarchy of this game, and we also give a lower bound of Ω(logd)\Omega(\log d) for its strong price of anarchy

    Complexity Theory, Game Theory, and Economics: The Barbados Lectures

    Full text link
    This document collects the lecture notes from my mini-course "Complexity Theory, Game Theory, and Economics," taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity. The goal of this mini-course is twofold: (i) to explain how complexity theory has helped illuminate several barriers in economics and game theory; and (ii) to illustrate how game-theoretic questions have led to new and interesting complexity theory, including recent several breakthroughs. It consists of two five-lecture sequences: the Solar Lectures, focusing on the communication and computational complexity of computing equilibria; and the Lunar Lectures, focusing on applications of complexity theory in game theory and economics. No background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some recent citations to v1 Revised v3 corrects a few typos in v

    Collocation Games and Their Application to Distributed Resource Management

    Full text link
    We introduce Collocation Games as the basis of a general framework for modeling, analyzing, and facilitating the interactions between the various stakeholders in distributed systems in general, and in cloud computing environments in particular. Cloud computing enables fixed-capacity (processing, communication, and storage) resources to be offered by infrastructure providers as commodities for sale at a fixed cost in an open marketplace to independent, rational parties (players) interested in setting up their own applications over the Internet. Virtualization technologies enable the partitioning of such fixed-capacity resources so as to allow each player to dynamically acquire appropriate fractions of the resources for unencumbered use. In such a paradigm, the resource management problem reduces to that of partitioning the entire set of applications (players) into subsets, each of which is assigned to fixed-capacity cloud resources. If the infrastructure and the various applications are under a single administrative domain, this partitioning reduces to an optimization problem whose objective is to minimize the overall deployment cost. In a marketplace, in which the infrastructure provider is interested in maximizing its own profit, and in which each player is interested in minimizing its own cost, it should be evident that a global optimization is precisely the wrong framework. Rather, in this paper we use a game-theoretic framework in which the assignment of players to fixed-capacity resources is the outcome of a strategic "Collocation Game". Although we show that determining the existence of an equilibrium for collocation games in general is NP-hard, we present a number of simplified, practically-motivated variants of the collocation game for which we establish convergence to a Nash Equilibrium, and for which we derive convergence and price of anarchy bounds. In addition to these analytical results, we present an experimental evaluation of implementations of some of these variants for cloud infrastructures consisting of a collection of multidimensional resources of homogeneous or heterogeneous capacities. Experimental results using trace-driven simulations and synthetically generated datasets corroborate our analytical results and also illustrate how collocation games offer a feasible distributed resource management alternative for autonomic/self-organizing systems, in which the adoption of a global optimization approach (centralized or distributed) would be neither practical nor justifiable.NSF (CCF-0820138, CSR-0720604, EFRI-0735974, CNS-0524477, CNS-052016, CCR-0635102); Universidad Pontificia Bolivariana; COLCIENCIAS–Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnología "Francisco José de Caldas

    Embedding Games

    Full text link
    Large scale distributed computing infrastructures pose challenging resource management problems, which could be addressed by adopting one of two perspectives. On the one hand, the problem could be framed as a global optimization that aims to minimize some notion of system-wide (social) cost. On the other hand, the problem could be framed in a game-theoretic setting whereby rational, selfish users compete for a share of the resources so as to maximize their private utilities with little or no regard for system-wide objectives. This game-theoretic setting is particularly applicable to emerging cloud and grid environments, testbed platforms, and many networking applications. By adopting the first, global optimization perspective, this thesis presents NetEmbed: a framework, associated mechanisms, and implementations that enable the mapping of requested configurations to available infrastructure resources. By adopting the second, game-theoretic perspective, this thesis defines and establishes the premises of two resource acquisition mechanisms: Colocation Games and Trade and Cap. Colocation Games enable the modeling and analysis of the dynamics that result when rational, selfish parties interact in an attempt to minimize the individual costs they incur to secure shared resources necessary to support their application QoS or SLA requirements. Trade and Cap is a market-based scheduling and load-balancing mechanism that facilitates the trading of resources when users have a mixture of rigid and fluid jobs, and incentivizes users to behave in ways that result in better load-balancing of shared resources. In addition to developing their analytical underpinnings, this thesis establishes the viability of NetEmbed, Colocation Games, and Trade and Cap by presenting implementation blueprints and experimental results for many variants of these mechanisms. The results presented in this thesis pave the way for the development of economically-sound resource acquisition and management solutions in two emerging, and increasingly important settings. In pay-as-you-go settings, where pricing is based on usage, this thesis anticipates new service offerings that enable efficient marketplaces in the presence of non-cooperative, selfish agents. In settings where pricing is not a function of usage, this thesis anticipates the development of service offerings that enable trading of usage rights to maximize the utility of a shared infrastructure to its tenants

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    27th Annual European Symposium on Algorithms: ESA 2019, September 9-11, 2019, Munich/Garching, Germany

    Get PDF

    A tale of two packing problems : improved algorithms and tighter bounds for online bin packing and the geometric knapsack problem

    Get PDF
    In this thesis, we deal with two packing problems: the online bin packing and the geometric knapsack problem. In online bin packing, the aim is to pack a given number of items of different size into a minimal number of containers. The items need to be packed one by one without knowing future items. For online bin packing in one dimension, we present a new family of algorithms that constitutes the first improvement over the previously best algorithm in almost 15 years. While the algorithmic ideas are intuitive, an elaborate analysis is required to prove its competitive ratio. We also give a lower bound for the competitive ratio of this family of algorithms. For online bin packing in higher dimensions, we discuss lower bounds for the competitive ratio and show that the ideas from the one-dimensional case cannot be easily transferred to obtain better two-dimensional algorithms. In the geometric knapsack problem, one aims to pack a maximum weight subset of given rectangles into one square container. For this problem, we consider online approximation algorithms. For geometric knapsack with square items, we improve the running time of the best known PTAS and obtain an EPTAS. This shows that large running times caused by some standard techniques for geometric packing problems are not always necessary and can be improved. Finally, we show how to use resource augmentation to compute optimal solutions in EPTAS-time, thereby improving upon the known PTAS for this case.In dieser Arbeit betrachten wir zwei Packungsprobleme: Online Bin Packing und das geometrische Rucksackproblem. Bei Online Bin Packing versucht man, eine gegebene Menge an Objekten verschiedener Größe in die kleinstmögliche Anzahl an Behältern zu packen. Die Objekte müssen eins nach dem anderen gepackt werden, ohne zukünftige Objekte zu kennen. Für eindimensionales Online Bin Packing beschreiben wir einen neuen Algorithmus, der die erste Verbesserung gegenüber dem bisher besten Algorithmus seit fast 15 Jahren darstellt. Während die algorithmischen Ideen intuitiv sind, ist eine ausgefeilte Analyse notwendig um das Kompetitivitätsverhältnis zu beweisen. Für Online Bin Packing in mehreren Dimensionen geben wir untere Schranken für das Kompetitivitätsverhältnis an und zeigen, dass die Ideen aus dem eindimensionalen Fall nicht direkt zu einer Verbesserung führen. Beim geometrischen Rucksackproblem ist es das Ziel, eine größtmögliche Teilmenge gegebener Rechtecke in einen einzelnen quadratischen Behälter zu packen. Für dieses Problem betrachten wir Approximationsalgorithmen. Für das Problem mit quadratischen Objekten verbessern wir die Laufzeit des bekannten PTAS zu einem EPTAS. Die langen Laufzeiten vieler Standardtechniken für geometrische Probleme können also vermieden werden. Schließlich zeigen wir, wie Ressourcenvergrößerung genutzt werden kann, um eine optimale Lösung in EPTAS-Zeit zu berechnen, was das bisherige PTAS verbessert.Google PhD Fellowshi
    corecore