3 research outputs found
The Containment Problem for Unambiguous Register Automata
We investigate the complexity of the containment problem "Does L(A)subseteq L(B) hold?", where B is an unambiguous register automaton and A is an arbitrary register automaton. We prove that the problem is decidable and give upper bounds on the computational complexity in the general case, and when B is restricted to have a fixed number of registers
Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs
Parity games form an intriguing family of infinitary payoff games whose solution
is equivalent to the solution of important problems in automatic verification and
automata theory. They also form a very natural subclass of mean and discounted
payoff games, which in turn are very natural subclasses of turn-based stochastic
payoff games. From a theoretical point of view, solving these games is one of the few
problems that belong to the complexity class NP intersect coNP, and even more interestingly,
solving has been shown to belong to UP intersect coUP, and also to PLS. It is a major open
problem whether these game families can be solved in deterministic polynomial
time.
Policy iteration is one of the most important algorithmic schemes for solving
infinitary payoff games. It is parameterized by an improvement rule that determines
how to proceed in the iteration from one policy to the next. It is a major open problem
whether there is an improvement rule that results in a polynomial time algorithm for
solving one of the considered game classes.
Linear programming is one of the most important computational problems studied
by researchers in computer science, mathematics and operations research. Perhaps
more articles and books are written about linear programming than on all other
computational problems combined.
The simplex and the dual-simplex algorithms are among the most widely used
algorithms for solving linear programs in practice. Simplex algorithms for solving
linear programs are closely related to policy iteration algorithms. Like policy iteration,
the simplex algorithm is parameterized by a pivoting rule that describes how
to proceed from one basic feasible solution in the linear program to the next. It is
a major open problem whether there is a pivoting rule that results in a (strongly)
polynomial time algorithm for solving linear programs.
We contribute to both the policy iteration and the simplex algorithm by proving
exponential lower bounds for several improvement resp. pivoting rules. For every
considered improvement rule, we start by building 2-player parity games on which
the respective policy iteration algorithm performs an exponential number of iterations.
We then transform these 2-player games into 1-player Markov decision processes
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which correspond almost immediately to concrete linear programs on which the
respective simplex algorithm requires the same number of iterations. Additionally,
we show how to transfer the lower bound results to more expressive game classes
like payoff and turn-based stochastic games.
Particularly, we prove exponential lower bounds for the deterministic switch
all and switch best improvement rules for solving games, for which no non-trivial
lower bounds have been known since the introduction of Howard’s policy iteration
algorithm in 1960. Moreover, we prove exponential lower bounds for the two most
natural and most studied randomized pivoting rules suggested to date, namely the random
facet and random edge rules for solving games and linear programs, for which
no non-trivial lower bounds have been known for several decades. Furthermore, we
prove an exponential lower bound for the switch half randomized improvement rule
for solving games, which is considered to be the most important multi-switching
randomized rule. Finally, we prove an exponential lower bound for the most natural
and famous history-based pivoting rule due to Zadeh for solving games and linear
programs, which has been an open problem for thirty years.
Last but not least, we prove exponential lower bounds for two other classes of
algorithms that solve parity games, namely for the model checking algorithm due to
Stevens and Stirling and for the recursive algorithm by Zielonka