Succinct progress measures for solving parity games

The recent breakthrough paper by Calude et al. has given the first algorithm for solving parity games in quasi-polynomial time, where previously the best algorithms were mildly subexponential. We devise an alternative quasi-polynomial time algorithm based on progress measures, which allows us to reduce the space required from quasi-polynomial to nearly linear. Our key technical tools are a novel concept of ordered tree coding, and a succinct tree coding result that we prove using bounded adaptive multi-counters, both of which are interesting in their own right

Alternating weak automata from universal trees

An improved translation from alternating parity automata on infinite words to alternating weak automata is given. The blow-up of the number of states is related to the size of the smallest universal ordered trees and hence it is quasi-polynomial, and it is polynomial if the asymptotic number of priorities is at most logarithmic in the number of states. This is an exponential improvement on the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on the translation of Boker and Lehtinen (2018). Any slightly better such translation would (if - like all presently known such translations - it is efficiently constructive) lead to algorithms for solving parity games that are asymptotically faster in the worst case than the current state of the art (Calude, Jain, Khoussainov, Li, and Stephan, 2017; Jurdzinski and Lazic, 2017; and Fearnley, Jain, Schewe, Stephan, and Wojtczak, 2017), and hence it would yield a significant breakthrough

Perfect half space games

We introduce perfect half space games, in which the goal of Player 2 is to make the sums of encountered multidimensional weights diverge in a direction which is consistent with a chosen sequence of perfect half spaces (chosen dynamically by Player 2). We establish that the bounding games of Jurdzinski et al. (ICALP 2015) can be reduced to perfect half space games, which in turn can be translated to the lexicographic energy games of Colcombet and Niwinski, and are positionally determined in a strong sense (Player 2 can play without knowing the current perfect half space). We finally show how perfect half space games and bounding games can be employed to solve multidimensional energy parity games in pseudo-polynomial time when both the numbers of energy dimensions and of priorities are fixed, regardless of whether the initial credit is given as part of the input or existentially quantified. This also yields an optimal 2-EXPTIME complexity with given initial credit, where the best known upper bound was non-elementary

Mean-Payoff Games on Timed Automata

Mean-payoff games on timed automata are played on the infinite weighted graph of configurations of priced timed automata between two players - Player Min and Player Max - by moving a token along the states of the graph to form an infinite run. The goal of Player Min is to minimize the limit average weight of the run, while the goal of the Player Max is the opposite. Brenguier, Cassez, and Raskin recently studied a variation of these games and showed that mean-payoff games are undecidable for timed automata with five or more clocks. We refine this result by proving the undecidability of mean-payoff games with three clocks. On a positive side, we show the decidability of mean-payoff games on one-clock timed automata with binary price-rates. A key contribution of this paper is the application of dynamic programming based proof techniques applied in the context of average reward optimization on an uncountable state and action space

Sets Which Contain a Quadratic Residue Modulo p for Almost All p

In a mean-payoff parity game, one of the two players aims both to achieve a qualitative parity objective and to minimize a quantitative long-term average of payoffs (aka. mean payoff). The game is zero-sum and hence the aim of the other player is to either foil the parity objective or to maximize the mean payoff. Our main technical result is a pseudo-quasi-polynomial algorithm for solving mean-payoff parity games. All algorithms for the problem that have been developed for over a decade have a pseudo-polynomial and an exponential factors in their running times; in the running time of our algorithm the latter is replaced with a quasi-polynomial one. By the results of Chatterjee and Doyen (2012) and of Schewe, Weinert, and Zimmermann (2018), our main technical result implies that there are pseudo-quasi-polynomial algorithms for solving parity energy games and for solving parity games with weights. Our main conceptual contributions are the definitions of strategy decompositions for both players, and a notion of progress measures for mean-payoff parity games that generalizes both parity and energy progress measures. The former provides normal forms for and succinct representations of winning strategies, and the latter enables the application to mean-payoff parity games of the order-theoretic machinery that underpins a recent quasi-polynomial algorithm for solving parity games

Distributed Online and Stochastic Queueing on a Multiple Access Channel

We consider the problems of online and stochastic packet queueing in a distributed system of n nodes with queues, where the communication between the nodes is done via a multiple access channel. In the online setting, in each round, an arbitrary number of packets can be injected to nodes’ queues. Two measures of performance are considered: the total number of packets in all queues, called the total load , and the maximum queue size, called the maximum load . We develop a deterministic distributed algorithm that is asymptotically optimal with respect to both complexity measures, in a competitive way. More precisely, the total load of our algorithm is bigger than the total load of any other algorithm, including centralized online solutions, by only an additive term of O ( n 2 ), whereas the maximum queue size of our algorithm is at most n times bigger than the maximum queue size of any other algorithm, with an extra additive O ( n ). The optimality for both measures is justified by proving the corresponding lower bounds, which also separates nearly exponentially distributed solutions from the centralized ones. Next, we show that our algorithm is also stochastically stable for any expected injection rate smaller or equal to 1. This is the first solution to the stochastic queueing problem on a multiple access channel that achieves such stability for the (highest possible) rate equal to 1. </jats:p

Stochastic Timed Automata

A stochastic timed automaton is a purely stochastic process defined on a timed automaton, in which both delays and discrete choices are made randomly. We study the almost-sure model-checking problem for this model, that is, given a stochastic timed automaton A and a property $\Phi$, we want to decide whether A satisfies $\Phi$ with probability 1. In this paper, we identify several classes of automata and of properties for which this can be decided. The proof relies on the construction of a finite abstraction, called the thick graph, that we interpret as a finite Markov chain, and for which we can decide the almost-sure model-checking problem. Correctness of the abstraction holds when automata are almost-surely fair, which we show, is the case for two large classes of systems, single- clock automata and so-called weak-reactive automata. Techniques employed in this article gather tools from real-time verification and probabilistic verification, as well as topological games played on timed automata.Comment: 40 pages + appendi

Orthogonality in the Category of Complexes

Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register games. We argue that all those techniques can be viewed as instances of the separation approach to solving parity games, a key technical component of which is constructing (explicitly or implicitly) an automaton that separates languages of words encoding plays that are (decisively) won by either of the two players. Our main technical result is a quasi-polynomial lower bound on the size of such separating automata that nearly matches the current best upper bounds. This forms a barrier that all existing approaches must overcome in the ongoing quest for a polynomial-time algorithm for solving parity games. The key and fundamental concept that we introduce and study is a universal ordered tree. The technical highlights are a quasi-polynomial lower bound on the size of universal ordered trees and a proof that every separating safety automaton has a universal tree hidden in its state space

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 computes an approximate Nash equilibrium using only limited communication between the players. Our method gives improved bounds on the complexity of computing approximate Nash equilibria in a number of different settings. Firstly, it gives a polynomial-time algorithm for computing approximate well supported Nash equilibria (WSNE) that always finds a 0.6528-WSNE, beating the previous best guarantee of 0.6608. Secondly, since our algorithm solves the two LPs separately, it can be applied to give an improved bound in the limited communication setting, giving a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE, which beats the previous best known guarantee of 0.732. It can also be applied to the case of approximate Nash equilibria, where we obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and always finds a 0.382-approximate Nash equilibrium, which improves the previous best guarantee of 0.438. Finally, the method can also be applied in the query complexity setting to give an algorithm that makes O(nlogn) payoff queries and always finds a 0.6528-WSNE, which improves the previous best known guarantee of 2/3