124 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
Packing, Scheduling and Covering Problems in a Game-Theoretic Perspective
Many packing, scheduling and covering problems that were previously
considered by computer science literature in the context of various
transportation and production problems, appear also suitable for describing and
modeling various fundamental aspects in networks optimization such as routing,
resource allocation, congestion control, etc. Various combinatorial problems
were already studied from the game theoretic standpoint, and we attempt to
complement to this body of research.
Specifically, we consider the bin packing problem both in the classic and
parametric versions, the job scheduling problem and the machine covering
problem in various machine models. We suggest new interpretations of such
problems in the context of modern networks and study these problems from a game
theoretic perspective by modeling them as games, and then concerning various
game theoretic concepts in these games by combining tools from game theory and
the traditional combinatorial optimization. In the framework of this research
we introduce and study models that were not considered before, and also improve
upon previously known results.Comment: PhD thesi
Approximation algorithms for distributed and selfish agents
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 157-165).Many real-world systems involve distributed and selfish agents who optimize their own objective function. In these systems, we need to design efficient mechanisms so that system-wide objective is optimized despite agents acting in their own self interest. In this thesis, we develop approximation algorithms and decentralized mechanisms for various combinatorial optimization problems in such systems. First, we investigate the distributed caching and a general set of assignment problems. We develop an almost tight LP-based ... approximation algorithm and a local search ... approximation algorithm for these problems. We also design efficient decentralized mechanisms for these problems and study the convergence of the corresponding games. In the following chapters, we study the speed of convergence to high quality solutions on (random) best-response paths of players. First, we study the average social value on best response paths in basic-utility, market sharing, and cut games. Then, we introduce the sink equilibrium as a new equilibrium concept. We argue that, unlike Nash equilibria, the selfish behavior of players converges to sink equilibria and all strategic games have a sink equilibrium. To illustrate the use of this new concept, we study the social value of sink equilibria in weighted selfish routing (or weighted congestion) games and valid-utility (or submodular-utility) games. In these games, we bound the average social value on random best-response paths for sink equilibria.. Finally, we study cross-monotonic cost sharings and group-strategyproof mechanisms.(cont.) We study the limitations imposed by the cross-monotonicity property on cost-sharing schemes for several combinatorial optimization games including set cover and metric facility location. We develop a novel technique based on the probabilistic method for proving upper bounds on the budget-balance factor of cross-monotonic cost sharing schemes, deriving tight or nearly-tight bounds for these games. At the end, we extend some of these results to group-strategyproof mechanisms.by Vahab S. Mirrokni.Ph.D
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
Incentives in dynamic markets
In this thesis, we consider a variety of combinatorial optimization problems within a common theme of uncertainty and selfish behavior. In our first scenario, the input is collected from selfish players. Here, we study extensions of the so-called smoothness framework for mechanisms, a very useful technique for bounding the inefficiency of equilibria, to the cases of varying mechanism availability and participation of risk-averse players. In both of these cases, our main results are general theorems for the class of (lambda,mu)-smooth mechanisms. We show that these mechanisms guarantee at most a (small) constant factor performance loss in the extended settings. In our second scenario, we do not have access to the exact numerical input. Within this context, we explore combinatorial extensions of the well-known secretary problem under the assumption that the incoming elements only reveal their ordinal position within the set of previously arrived elements. We first observe that many existing algorithms for special matroid structures maintain their competitive ratio in the ordinal model. In contrast, we provide a lower bound for algorithms that are oblivious to the matroid structure. Finally, we design new algorithms that obtain constant competitive ratios for a variety of combinatorial problems
Portland Daily Press: May 19,1886
https://digitalmaine.com/pdp_1886/1117/thumbnail.jp
Albuquerque Morning Journal, 04-16-1915
https://digitalrepository.unm.edu/abq_mj_news/2266/thumbnail.jp
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