22 research outputs found
Violator Spaces: Structure and Algorithms
Sharir and Welzl introduced an abstract framework for optimization problems,
called LP-type problems or also generalized linear programming problems, which
proved useful in algorithm design. We define a new, and as we believe, simpler
and more natural framework: violator spaces, which constitute a proper
generalization of LP-type problems. We show that Clarkson's randomized
algorithms for low-dimensional linear programming work in the context of
violator spaces. For example, in this way we obtain the fastest known algorithm
for the P-matrix generalized linear complementarity problem with a constant
number of blocks. We also give two new characterizations of LP-type problems:
they are equivalent to acyclic violator spaces, as well as to concrete LP-type
problems (informally, the constraints in a concrete LP-type problem are subsets
of a linearly ordered ground set, and the value of a set of constraints is the
minimum of its intersection).Comment: 28 pages, 5 figures, extended abstract was presented at ESA 2006;
author spelling fixe
Policy iteration algorithm for zero-sum stochastic games with mean payoff
We give a policy iteration algorithm to solve zero-sum stochastic games with finite state and action spaces and perfect information, when the value is defined in terms of the mean payoff per turn. This algorithm does not require any irreducibility assumption on the Markov chains determined by the strategies of the players. It is based on a discrete nonlinear analogue of the notion of reduction of a super-harmonic function
Non-oblivious Strategy Improvement
We study strategy improvement algorithms for mean-payoff and parity games. We
describe a structural property of these games, and we show that these
structures can affect the behaviour of strategy improvement. We show how
awareness of these structures can be used to accelerate strategy improvement
algorithms. We call our algorithms non-oblivious because they remember
properties of the game that they have discovered in previous iterations. We
show that non-oblivious strategy improvement algorithms perform well on
examples that are known to be hard for oblivious strategy improvement. Hence,
we argue that previous strategy improvement algorithms fail because they ignore
the structural properties of the game that they are solving
Oink: an Implementation and Evaluation of Modern Parity Game Solvers
Parity games have important practical applications in formal verification and
synthesis, especially to solve the model-checking problem of the modal
mu-calculus. They are also interesting from the theory perspective, as they are
widely believed to admit a polynomial solution, but so far no such algorithm is
known. In recent years, a number of new algorithms and improvements to existing
algorithms have been proposed. We implement a new and easy to extend tool Oink,
which is a high-performance implementation of modern parity game algorithms. We
further present a comprehensive empirical evaluation of modern parity game
algorithms and solvers, both on real world benchmarks and randomly generated
games. Our experiments show that our new tool Oink outperforms the current
state-of-the-art.Comment: Accepted at TACAS 201
Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games
2.5 player parity games combine the challenges posed by 2.5 player
reachability games and the qualitative analysis of parity games. These two
types of problems are best approached with different types of algorithms:
strategy improvement algorithms for 2.5 player reachability games and recursive
algorithms for the qualitative analysis of parity games. We present a method
that - in contrast to existing techniques - tackles both aspects with the best
suited approach and works exclusively on the 2.5 player game itself. The
resulting technique is powerful enough to handle games with several million
states
Tropical Fourier-Motzkin elimination, with an application to real-time verification
We introduce a generalization of tropical polyhedra able to express both
strict and non-strict inequalities. Such inequalities are handled by means of a
semiring of germs (encoding infinitesimal perturbations). We develop a tropical
analogue of Fourier-Motzkin elimination from which we derive geometrical
properties of these polyhedra. In particular, we show that they coincide with
the tropically convex union of (non-necessarily closed) cells that are convex
both classically and tropically. We also prove that the redundant inequalities
produced when performing successive elimination steps can be dynamically
deleted by reduction to mean payoff game problems. As a complement, we provide
a coarser (polynomial time) deletion procedure which is enough to arrive at a
simply exponential bound for the total execution time. These algorithms are
illustrated by an application to real-time systems (reachability analysis of
timed automata).Comment: 29 pages, 8 figure