29 research outputs found
Fixed-Dimensional Energy Games are in Pseudo-Polynomial Time
We generalise the hyperplane separation technique (Chatterjee and Velner,
2013) from multi-dimensional mean-payoff to energy games, and achieve an
algorithm for solving the latter whose running time is exponential only in the
dimension, but not in the number of vertices of the game graph. This answers an
open question whether energy games with arbitrary initial credit can be solved
in pseudo-polynomial time for fixed dimensions 3 or larger (Chaloupka, 2013).
It also improves the complexity of solving multi-dimensional energy games with
given initial credit from non-elementary (Br\'azdil, Jan\v{c}ar, and
Ku\v{c}era, 2010) to 2EXPTIME, thus establishing their 2EXPTIME-completeness.Comment: Corrected proof of Lemma 6.2 (thanks to Dmitry Chistikov for spotting
an error in the previous proof
A Universal Attractor Decomposition Algorithm for Parity Games
An attractor decomposition meta-algorithm for solving parity games is given
that generalizes the classic McNaughton-Zielonka algorithm and its recent
quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and
Wojtczak (2019). The central concepts studied and exploited are attractor
decompositions of dominia in parity games and the ordered trees that describe
the inductive structure of attractor decompositions.
The main technical results include the embeddable decomposition theorem and
the dominion separation theorem that together help establish a precise
structural condition for the correctness of the universal algorithm: it
suffices that the two ordered trees given to the algorithm as inputs embed the
trees of some attractor decompositions of the largest dominia for each of the
two players, respectively.
The universal algorithm yields McNaughton-Zielonka, Parys's, and
Lehtinen-Schewe-Wojtczak algorithms as special cases when suitable universal
trees are given to it as inputs. The main technical results provide a unified
proof of correctness and deep structural insights into those algorithms.
A symbolic implementation of the universal algorithm is also given that
improves the symbolic space complexity of solving parity games in
quasi-polynomial time from ---achieved by Chatterjee,
Dvo\v{r}\'{a}k, Henzinger, and Svozil (2018)---down to , where is
the number of vertices and is the number of distinct priorities in a parity
game. This not only exponentially improves the dependence on , but it also
entirely removes the dependence on
Distributed Methods for Computing Approximate Equilibria
We present a new, distributed method to compute approximate Nash equilibria
in bimatrix games. In contrast to previous approaches that analyze the two
payoff matrices at the same time (for example, by solving a single LP that
combines the two players payoffs), our algorithm first solves two independent
LPs, each of which is derived from one of the two payoff matrices, and then
compute approximate Nash equilibria using only limited communication between
the players.
Our method has several applications for improved bounds for efficient
computations of approximate Nash equilibria in bimatrix games. First, it yields
a best polynomial-time algorithm for computing \emph{approximate well-supported
Nash equilibria (WSNE)}, which guarantees to find a 0.6528-WSNE in polynomial
time. Furthermore, since our algorithm solves the two LPs separately, it can be
used to improve upon the best known algorithms in the limited communication
setting: the algorithm can be implemented to obtain a randomized
expected-polynomial-time algorithm that uses poly-logarithmic communication and
finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best
known bound in the query complexity setting, requiring payoff
queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to
provide the best known communication efficient algorithm for computing
\emph{approximate Nash equilibria}: it uses poly-logarithmic communication to
find a 0.382-approximate Nash equilibrium
Games for Verification: Algorithmic Issues
This dissertation deals with a number of algorithmic problems motivated by computer aided formal verification of finite state systems. The goal of formal verification is to enhance the design and development of complex systems by providing methods and tools for specifying and verifying correctness of designs. The success of formal methods in practice depends heavily on the degree of automation of development and verification process. This motivates development of efficient algorithms for problems underlying many verification tasks. Tw
This document in subdirectoryRS/99/19/ Hereditary History Preserving Bisimilarity Is Undecidable
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This document in subdirectoryRS/99/1/ Hereditary History Preserving Simulation Is Undecidable
Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRIC
Reachability-time games on timed automata
Abstract. In a reachability-time game, players Min and Max choose moves so that the time to reach a final state in a timed automaton is minimised or maximised, respectively. Asarin and Maler showed decidability of reachability-time games on strongly non-Zeno timed automata using a value iteration algorithm. This paper complements their work by providing a strategy improvement algorithm for the problem. It also generalizes their decidability result because the proposed strategy improvement algorithm solves reachability-time games on all timed automata. The exact computational complexity of solving reachability-time games is also established: the problem is EXPTIME-complete for timed automata with at least two clocks.