7,625 research outputs found
Presburger arithmetic, rational generating functions, and quasi-polynomials
Presburger arithmetic is the first-order theory of the natural numbers with
addition (but no multiplication). We characterize sets that can be defined by a
Presburger formula as exactly the sets whose characteristic functions can be
represented by rational generating functions; a geometric characterization of
such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the
free variables in a Presburger formula, we can define a counting function g(p)
to be the number of solutions to the formula, for a given p. We show that every
counting function obtained in this way may be represented as, equivalently,
either a piecewise quasi-polynomial or a rational generating function. Finally,
we translate known computational complexity results into this setting and
discuss open directions.Comment: revised, including significant additions explaining computational
complexity results. To appear in Journal of Symbolic Logic. Extended abstract
in ICALP 2013. 17 page
A parametric integer programming algorithm for bilevel mixed integer programs
We consider discrete bilevel optimization problems where the follower solves
an integer program with a fixed number of variables. Using recent results in
parametric integer programming, we present polynomial time algorithms for pure
and mixed integer bilevel problems. For the mixed integer case where the
leader's variables are continuous, our algorithm also detects whether the
infimum cost fails to be attained, a difficulty that has been identified but
not directly addressed in the literature. In this case it yields a ``better
than fully polynomial time'' approximation scheme with running time polynomial
in the logarithm of the relative precision. For the pure integer case where the
leader's variables are integer, and hence optimal solutions are guaranteed to
exist, we present two algorithms which run in polynomial time when the total
number of variables is fixed.Comment: 11 page
On minimum sum representations for weighted voting games
A proposal in a weighted voting game is accepted if the sum of the
(non-negative) weights of the "yea" voters is at least as large as a given
quota. Several authors have considered representations of weighted voting games
with minimum sum, where the weights and the quota are restricted to be
integers. Freixas and Molinero have classified all weighted voting games
without a unique minimum sum representation for up to 8 voters. Here we
exhaustively classify all weighted voting games consisting of 9 voters which do
not admit a unique minimum sum integer weight representation.Comment: 7 pages, 6 tables; enumerations correcte
An Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games
This paper presents a technique for approximating, up to any precision, the
set of subgame-perfect equilibria (SPE) in discounted repeated games. The
process starts with a single hypercube approximation of the set of SPE. Then
the initial hypercube is gradually partitioned on to a set of smaller adjacent
hypercubes, while those hypercubes that cannot contain any point belonging to
the set of SPE are simultaneously withdrawn.
Whether a given hypercube can contain an equilibrium point is verified by an
appropriate mathematical program. Three different formulations of the algorithm
for both approximately computing the set of SPE payoffs and extracting players'
strategies are then proposed: the first two that do not assume the presence of
an external coordination between players, and the third one that assumes a
certain level of coordination during game play for convexifying the set of
continuation payoffs after any repeated game history.
A special attention is paid to the question of extracting players' strategies
and their representability in form of finite automata, an important feature for
artificial agent systems.Comment: 26 pages, 13 figures, 1 tabl
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