5,597 research outputs found
Exact Algorithms for Solving Stochastic Games
Shapley's discounted stochastic games, Everett's recursive games and
Gillette's undiscounted stochastic games are classical models of game theory
describing two-player zero-sum games of potentially infinite duration. We
describe algorithms for exactly solving these games
Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
A general issue in computational optimization is to develop combinatorial
algorithms for semidefinite programming. We address this issue when the base
field is nonarchimedean. We provide a solution for a class of semidefinite
feasibility problems given by generic matrices. Our approach is based on
tropical geometry. It relies on tropical spectrahedra, which are defined as the
images by the valuation of nonarchimedean spectrahedra. We establish a
correspondence between generic tropical spectrahedra and zero-sum stochastic
games with perfect information. The latter have been well studied in
algorithmic game theory. This allows us to solve nonarchimedean semidefinite
feasibility problems using algorithms for stochastic games. These algorithms
are of a combinatorial nature and work for large instances.Comment: v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions +
benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign
patterns + some results have been replaced by references to the companion
work arXiv:1610.0674
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
Robust Coin Flipping
Alice seeks an information-theoretically secure source of private random
data. Unfortunately, she lacks a personal source and must use remote sources
controlled by other parties. Alice wants to simulate a coin flip of specified
bias , as a function of data she receives from sources; she seeks
privacy from any coalition of of them. We show: If , the
bias can be any rational number and nothing else; if , the bias
can be any algebraic number and nothing else. The proof uses projective
varieties, convex geometry, and the probabilistic method. Our results improve
on those laid out by Yao, who asserts one direction of the case in his
seminal paper [Yao82]. We also provide an application to secure multiparty
computation.Comment: 22 pages, 1 figur
Tropically convex constraint satisfaction
A semilinear relation S is max-closed if it is preserved by taking the
componentwise maximum. The constraint satisfaction problem for max-closed
semilinear constraints is at least as hard as determining the winner in Mean
Payoff Games, a notorious problem of open computational complexity. Mean Payoff
Games are known to be in the intersection of NP and co-NP, which is not known
for max-closed semilinear constraints. Semilinear relations that are max-closed
and additionally closed under translations have been called tropically convex
in the literature. One of our main results is a new duality for open tropically
convex relations, which puts the CSP for tropically convex semilinaer
constraints in general into NP intersected co-NP. This extends the
corresponding complexity result for scheduling under and-or precedence
constraints, or equivalently the max-atoms problem. To this end, we present a
characterization of max-closed semilinear relations in terms of syntactically
restricted first-order logic, and another characterization in terms of a finite
set of relations L that allow primitive positive definitions of all other
relations in the class. We also present a subclass of max-closed constraints
where the CSP is in P; this class generalizes the class of max-closed
constraints over finite domains, and the feasibility problem for max-closed
linear inequalities. Finally, we show that the class of max-closed semilinear
constraints is maximal in the sense that as soon as a single relation that is
not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure
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