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Exponential Number (exponential + number)

Distribution by Scientific Domains
Business, Economics, Finance and Accounting50%

Selected Abstracts

Hard-to-Solve Bimatrix Games

ECONOMETRICA, Issue 2 2006
Rahul Savani
The Lemke,Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d -space. The construction is extended to nonsquare games where, in addition to exponentially long Lemke,Howson computations, finding an equilibrium by support enumeration takes on average exponential time. [source]

Efficient Universal Portfolios for Past-Dependent Target Classes

MATHEMATICAL FINANCE, Issue 2 2003
Jason E. Cross
We present a new universal portfolio algorithm that achieves almost the same level of wealth as could be achieved by knowing stock prices ahead of time. Specifically the algorithm tracks the best in hindsight wealth achievable within target classes of linearly parameterized portfolio sequences. The target classes considered are more general than the standard constant rebalanced portfolio class and permit portfolio sequences to exhibit a continuous form of dependence on past prices or other side information. A primary advantage of the algorithm is that it is easily computable in a polynomial number of steps by way of simple closed-form expressions. This provides an edge over other universal algorithms that require both an exponential number of computations and numerical approximation. [source]

A Tractable and Expressive Class of Marginal Contribution Nets and Its Applications

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4 2009
Edith Elkind
Abstract Coalitional games raise a number of important questions from the point of view of computer science, key among them being how to represent such games compactly, and how to efficiently compute solution concepts assuming such representations. Marginal contribution nets (MC-nets), introduced by Ieong and Shoham, are one of the simplest and most influential representation schemes for coalitional games. MC-nets are a rulebased formalism, in which rules take the form pattern , value, where "pattern " is a Boolean condition over agents, and "value " is a numeric value. Ieong and Shoham showed that, for a class of what we will call "basic" MC-nets, where patterns are constrained to be a conjunction of literals, marginal contribution nets permit the easy computation of solution concepts such as the Shapley value. However, there are very natural classes of coalitional games that require an exponential number of such basic MC-net rules. We present read-once MC- nets, a new class of MC-nets that is provably more compact than basic MC-nets, while retaining the attractive computational properties of basic MC-nets. We show how the techniques we develop for read-once MC-nets can be applied to other domains, in particular, computing solution concepts in network flow games on series-parallel networks (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

A column generation approach for SONET ring assignment

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2006
Elder M. Macambira
Abstract In this article we consider the SONET ring assignment problem (SRAP) presented in 7. The authors pointed out the inadequacy of solving SRAP instances using their integer programming formulation and commercial linear programming solvers. Similar experiences with IP models for SRAP are reported in 1. In this article we reformulate SRAP as a set partitioning model with an additional knapsack constraint. This new formulation has an exponential number of columns and, to solve it, we implemented a branch-and-price/column generation algorithm. Extensive computational experiments showed that the new algorithm is orders of magnitude faster than standard branch-and-bound codes running on compact IP models introduced earlier. Instances taken from 1, 7, which could not be solved there in hours of computation were solved here to optimality in just a few seconds. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(3), 157,171 2006 [source]