22 research outputs found
A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game
We prove a tight lower bound on the asymptotic performance ratio of
the bounded space online -hypercube bin packing problem, solving an open
question raised in 2005. In the classic -hypercube bin packing problem, we
are given a sequence of -dimensional hypercubes and we have an unlimited
number of bins, each of which is a -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 -hypercube bin packing problem is a variant of the
-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 is and , and conjectured that
it is . We show that is in fact . 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 , using probabilistic arguments. Such packings also lead to
lower bounds for the prices of anarchy of the selfish -hypercube bin packing
game. We present a lower bound of for the pure price of
anarchy of this game, and we also give a lower bound of for
its strong price of anarchy
Complexity Theory, Game Theory, and Economics: The Barbados Lectures
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
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
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
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Recommended from our members
Budget Management in Auctions: Bidding Algorithms and Equilibrium Analysis
Advertising is the economic engine of the internet. It allows online platforms to fund services that are free at the point of use, while providing businesses the opportunity to target their ads at relevant users. The mechanism of choice for selling these advertising opportunities is real-time auctions: whenever a user visits the platform, an auction is run among interested advertisers, and the winner gets to display their ad to the user. The entire process runs in milliseconds and is implemented via automated algorithms which bid on behalf of the advertisers in every auction. These automated bidders take as input the high-level objectives of the advertiser like value-per-click and budget, and then participate in the auctions with the goal of maximizing the utility of the advertiser subject to budget constraints. Thus motivated, this thesis develops a theory of bidding in auctions under budget constraints, with the goal of informing the design of automated bidding algorithms and analyzing the market-level outcomes that emerge from their simultaneous use.
First, we take the perspective of an individual advertiser and tackle algorithm-design questions. How should one bid in repeated second-price auctions subject to a global budget constraint? What is the optimal way to incorporate data into bidding decisions? Can data be incorporated in a way that is robust to common forms of variability in the market? As we analyze these questions, we go beyond the problem of bidding under budget constraints and develop algorithms for more general online resource allocation problems. In Chapter 2, we study a non-stationary stochastic model of sequential auctions, which despite immense practical importance has received little attention, and propose a natural algorithm for it. With access to just one historical sample per auction/distribution, we show that our algorithm attains (nearly) the same performance as that possible under full knowledge of the distributions, while also being robust to distribution shifts which typically occur between the sampling and true distributions. Chapter 3 investigates the impact of uncertainty about the total number of auctions on the performance of bidding algorithms. We prove upper bounds on the best-possible performance that can be achieved in the face of such uncertainty, and propose an algorithm that (nearly) achieves this optimal performance guarantee. We also provide a fast method for incorporating predictions about the total number of auctions into our algorithm. All of our proposed algorithms implement some version of FTRL/Mirror-Descent in the dual space, making them ideal for large-scale low-latency markets like online advertising.
Next, we look at the market as a whole and analyze the equilibria which emerge from the simultaneous use of automated bidding algorithms. For example, we address questions like: Does an equilibrium always exist? How does the auction format (first-price vs second-price) impact the structure of the equilibria? Do automated bidding algorithms always efficiently converge to some equilibrium? What are the social welfare properties of these equilibrium outcomes? We systematically examine such questions using a variety of tools, ranging from infinite-dimensional fixed-point arguments for proving existence of structured equilibria, to computational complexity results about finding them. In Chapter 4, we start by establishing the existence of equilibria based on pacing—a practically-popular and theoretically-optimal budget management strategy—for all standard auctions, including first-price and second-price auctions. We then leverage its structure to establish a revenue equivalence result and bound the price of anarchy of liquid welfare. Chapter 5 looks at the market from a computational lens and investigates the complexity of finding pacing-based equilibria. We show that the problem is PPAD complete, which in turn implies the impossibility of polynomial-time convergence of any pacing-based automated bidding algorithms (under standard complexity-theoretic assumptions).
Finally, in Chapter 6, we move beyond pacing-based strategies and investigate throttling, which is another popular method for managing budgets in practice. Here, we describe a simple tâtonnement-style algorithm which efficiently converges to an equilibrium in first-price auctions, and show that no such algorithm exists for second-price auctions (under standard complexity-theoretic assumptions). Furthermore, we prove tight bounds on the price of anarchy for liquid welfare, and compare platform revenue under throttling and pacing
A tale of two packing problems : improved algorithms and tighter bounds for online bin packing and the geometric knapsack problem
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