arXiv subjects

Dominik Wojtczak

Publications and source records attributed to Dominik Wojtczak.

At least 37 records · Page 2Linked to original sources

arXiv2021

Coordination Games on Weighted Directed Graphs

We study strategic games on weighted directed graphs, in which the payoff of a player is defined as the sum of the weights on the edges from players who chose the same strategy, augmented by a fixed non-negative integer bonus for picking a given strategy. These games capture the idea of coordination in the absence of globally common strategies. We identify natural classes of graphs for which finite improvement or coalition-improvement paths of polynomial length always exist, and, as a consequence, a (pure) Nash equlibrium or a strong equilibrium can be found in polynomial time. The considered classes of graphs are typical in network topologies: simple cycles correspond to the token ring local area networks, while open chains of simple cycles correspond to multiple independent rings topology from the recommendation G.8032v2 on the Ethernet ring protection switching. For simple cycles these results are optimal in the sense that without the imposed conditions on the weights and bonuses a Nash equilibrium may not even exist. Finally, we prove that the problem of determining the existence of a Nash equilibrium or of a strong equilibrium in these games is NP-complete already for unweighted graphs and with no bonuses assumed. This implies that the same problems for polymatrix games are strongly NP-hard.

cs.GT↗

arXiv2021

Simple Stochastic Games with Almost-Sure Energy-Parity Objectives are in NP and coNP

We study stochastic games with energy-parity objectives, which combine quantitative rewards with a qualitative $ω$-regular condition: The maximizer aims to avoid running out of energy while simultaneously satisfying a parity condition. We show that the corresponding almost-sure problem, i.e., checking whether there exists a maximizer strategy that achieves the energy-parity objective with probability $1$ when starting at a given energy level $k$, is decidable and in $NP \cap coNP$. The same holds for checking if such a $k$ exists and if a given $k$ is minimal.

cs.GT↗

arXiv2021

Satisfiability Modulo Theories and Chiral Heterotic String Vacua with Positive Cosmological Constant

We apply Boolean Satisfiability (SAT) and Satisfiability Modulo Theories (SMT) solvers in the context of finding chiral heterotic string models with positive cosmological constant from $\mathbb{Z}_2\times \mathbb{Z}_2$ orbifolds. The power of using SAT/SMT solvers to sift large parameter spaces quickly to decide satisfiability, both to declare and prove unsatisfiability and to declare satisfiability, are demonstrated in this setting. These models are partly chosen to be small enough to plot the performance against exhaustive search, which takes around 2 hours 20 minutes to comb through the parameter space. We show that making use of SMT based techniques with integer encoding is rather simple and effective, while a more careful Boolean SAT encoding provides a significant speed-up -- determining satisfiability or unsatisfiability has, in our experiments varied between 0.03 and 0.06 seconds, while determining all models (where models exist) took 19 seconds for a constraint system that allows for 2048 models and 8.4 seconds for a constraint system that admits 640 models. We thus gain several orders of magnitude in speed, and this advantage is set to grow with a growing parameter space. This holds the promise that the method scales well beyond the initial problem we have used it for in this paper.

hep-th↗

arXiv2019

Good-for-MDPs Automata for Probabilistic Analysis and Reinforcement Learning

We characterize the class of nondeterministic $ω$-automata that can be used for the analysis of finite Markov decision processes (MDPs). We call these automata `good-for-MDPs' (GFM). We show that GFM automata are closed under classic simulation as well as under more powerful simulation relations that leverage properties of optimal control strategies for MDPs. This closure enables us to exploit state-space reduction techniques, such as those based on direct and delayed simulation, that guarantee simulation equivalence. We demonstrate the promise of GFM automata by defining a new class of automata with favorable properties - they are Büchi automata with low branching degree obtained through a simple construction - and show that going beyond limit-deterministic automata may significantly benefit reinforcement learning.

cs.FL↗

arXiv2019

Open Problems in a Logic of Gossips

Gossip protocols are programs used in a setting in which each agent holds a secret and the aim is to reach a situation in which all agents know all secrets. Such protocols rely on a point-to-point or group communication. Distributed epistemic gossip protocols use epistemic formulas in the component programs for the agents. The advantage of the use of epistemic logic is that the resulting protocols are very concise and amenable for a simple verification. Recently, we introduced a natural modal logic that allows one to express distributed epistemic gossip protocols and to reason about their correctness. We proved that the resulting protocols are implementable and that all aspects of their correctness, including termination, are decidable. To establish these results we showed that both the definition of semantics and of truth of the underlying logic are decidable. We also showed that the analogous results hold for an extension of this logic with the 'common knowledge' operator. However, several, often deceptively simple, questions about this logic and the corresponding gossip protocols remain open. The purpose of this paper is to list and elucidate these questions and provide for them an appropriate background information in the form of partial of related results.

cs.AI↗

arXiv2019

Improving the complexity of Parys' recursive algorithm

Parys has recently proposed a quasi-polynomial version of Zielonka's recursive algorithm for solving parity games. In this brief note we suggest a variation of his algorithm that improves the complexity to meet the state-of-the-art complexity of broadly $2^{O((\log n)(\log c))}$, while providing polynomial bounds when the number of colours is logarithmic.

cs.GT↗

arXiv2018

Omega-Regular Objectives in Model-Free Reinforcement Learning

We provide the first solution for model-free reinforcement learning of ω-regular objectives for Markov decision processes (MDPs). We present a constructive reduction from the almost-sure satisfaction of ω-regular objectives to an almost- sure reachability problem and extend this technique to learning how to control an unknown model so that the chance of satisfying the objective is maximized. A key feature of our technique is the compilation of ω-regular properties into limit- deterministic Buechi automata instead of the traditional Rabin automata; this choice sidesteps difficulties that have marred previous proposals. Our approach allows us to apply model-free, off-the-shelf reinforcement learning algorithms to compute optimal strategies from the observations of the MDP. We present an experimental evaluation of our technique on benchmark learning problems.

cs.LO↗

arXiv2018

On Strong NP-Completeness of Rational Problems

The computational complexity of the partition, 0-1 subset sum, unbounded subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple variants were studied in numerous papers in the past where all the weights and profits were assumed to be integers. We re-examine here the computational complexity of all these problems in the setting where the weights and profits are allowed to be any rational numbers. We show that all of these problems in this setting become strongly NP-complete and, as a result, no pseudo-polynomial algorithm can exist for solving them unless P=NP. Despite this result we show that they all still admit a fully polynomial-time approximation scheme.

cs.DM↗

arXiv2018

An Ordered Approach to Solving Parity Games in Quasi Polynomial Time and Quasi Linear Space

Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al.

cs.LO↗

arXiv2017

Common Knowledge in a Logic of Gossips

Gossip protocols aim at arriving, by means of point-to-point or group communications, at a situation in which all the agents know each other secrets. Recently a number of authors studied distributed epistemic gossip protocols. These protocols use as guards formulas from a simple epistemic logic, which makes their analysis and verification substantially easier. We study here common knowledge in the context of such a logic. First, we analyze when it can be reduced to iterated knowledge. Then we show that the semantics and truth for formulas without nested common knowledge operator are decidable. This implies that implementability, partial correctness and termination of distributed epistemic gossip protocols that use non-nested common knowledge operator is decidable, as well. Given that common knowledge is equivalent to an infinite conjunction of nested knowledge, these results are non-trivial generalizations of the corresponding decidability results for the original epistemic logic, established in (Apt & Wojtczak, 2016). K. R. Apt & D. Wojtczak (2016): On Decidability of a Logic of Gossips. In Proc. of JELIA 2016, pp. 18-33, doi:10.1007/ 978-3-319-48758-8_2.

cs.LO↗

arXiv2017

Optimal Control for Multi-Mode Systems with Discrete Costs

This paper studies optimal time-bounded control in multi-mode systems with discrete costs. Multi-mode systems are an important subclass of linear hybrid systems, in which there are no guards on transitions and all invariants are global. Each state has a continuous cost attached to it, which is linear in the sojourn time, while a discrete cost is attached to each transition taken. We show that an optimal control for this model can be computed in NEXPTIME and approximated in PSPACE. We also show that the one-dimensional case is simpler: although the problem is NP-complete (and in LOGSPACE for an infinite time horizon), we develop an FPTAS for finding an approximate solution.

cs.LO↗

arXiv2017

MDPs with Energy-Parity Objectives

Energy-parity objectives combine $ω$-regular with quantitative objectives of reward MDPs. The controller needs to avoid to run out of energy while satisfying a parity objective. We refute the common belief that, if an energy-parity objective holds almost-surely, then this can be realised by some finite memory strategy. We provide a surprisingly simple counterexample that only uses coBüchi conditions. We introduce the new class of bounded (energy) storage objectives that, when combined with parity objectives, preserve the finite memory property. Based on these, we show that almost-sure and limit-sure energy-parity objectives, as well as almost-sure and limit-sure storage parity objectives, are in $\mathit{NP}\cap \mathit{coNP}$ and can be solved in pseudo-polynomial time for energy-parity MDPs.

cs.LO↗

arXiv2017

Parity Objectives in Countable MDPs

We study countably infinite MDPs with parity objectives, and special cases with a bounded number of colors in the Mostowski hierarchy (including reachability, safety, Buchi and co-Buchi). In finite MDPs there always exist optimal memoryless deterministic (MD) strategies for parity objectives, but this does not generally hold for countably infinite MDPs. In particular, optimal strategies need not exist. For countable infinite MDPs, we provide a complete picture of the memory requirements of optimal (resp., $ε$-optimal) strategies for all objectives in the Mostowski hierarchy. In particular, there is a strong dichotomy between two different types of objectives. For the first type, optimal strategies, if they exist, can be chosen MD, while for the second type optimal strategies require infinite memory. (I.e., for all objectives in the Mostowski hierarchy, if finite-memory randomized strategies suffice then also MD strategies suffice.) Similarly, some objectives admit $ε$-optimal MD-strategies, while for others $ε$-optimal strategies require infinite memory. Such a dichotomy also holds for the subclass of countably infinite MDPs that are finitely branching, though more objectives admit MD-strategies here.

cs.LO↗

arXiv2017

On Strong Determinacy of Countable Stochastic Games

We study 2-player turn-based perfect-information stochastic games with countably infinite state space. The players aim at maximizing/minimizing the probability of a given event (i.e., measurable set of infinite plays), such as reachability, Büchi, omega-regular or more general objectives. These games are known to be weakly determined, i.e., they have value. However, strong determinacy of threshold objectives (given by an event and a threshold $c \in [0,1]$) was open in many cases: is it always the case that the maximizer or the minimizer has a winning strategy, i.e., one that enforces, against all strategies of the other player, that the objective is satisfied with probability $\ge c$ (resp. $< c$)? We show that almost-sure objectives (where $c=1$) are strongly determined. This vastly generalizes a previous result on finite games with almost-sure tail objectives. On the other hand we show that $\ge 1/2$ (co-)Büchi objectives are not strongly determined, not even if the game is finitely branching. Moreover, for almost-sure reachability and almost-sure Büchi objectives in finitely branching games, we strengthen strong determinacy by showing that one of the players must have a memoryless deterministic (MD) winning strategy.

cs.GT↗

arXiv2016

Constrained Pure Nash Equilibria in Polymatrix Games

We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs. The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy. These games capture the idea of coordination within a local neighbourhood in the absence of globally common strategies. We study the decision problem of checking whether a given set of strategy choices for a subset of the players is consistent with some pure Nash equilibrium or, alternatively, with all pure Nash equilibria. We identify the most natural tractable cases and show NP or coNP-completness of these problems already for unweighted DAGs.

cs.GT↗

arXiv2016

Coordination Games on Directed Graphs

We study natural strategic games on directed graphs, which capture the idea of coordination in the absence of globally common strategies. We show that these games do not need to have a pure Nash equilibrium and that the problem of determining their existence is NP-complete. The same holds for strong equilibria. We also exhibit some classes of games for which strong equilibria exist and prove that a strong equilibrium can then be found in linear time.

cs.GT↗

arXiv2016

Efficient Local Search in Coordination Games on Graphs

We study strategic games on weighted directed graphs, where the payoff of a player is defined as the sum of the weights on the edges from players who chose the same strategy augmented by a fixed non-negative bonus for picking a given strategy. These games capture the idea of coordination in the absence of globally common strategies. Prior work shows that the problem of determining the existence of a pure Nash equilibrium for these games is NP-complete already for graphs with all weights equal to one and no bonuses. However, for several classes of graphs (e.g. DAGs and cliques) pure Nash equilibria or even strong equilibria always exist and can be found by simply following a particular improvement or coalition-improvement path, respectively. In this paper we identify several natural classes of graphs for which a finite improvement or coalition-improvement path of polynomial length always exists, and, as a consequence, a Nash equilibrium or strong equilibrium in them can be found in polynomial time. We also argue that these results are optimal in the sense that in natural generalisations of these classes of graphs, a pure Nash equilibrium may not even exist.

cs.GT↗

arXiv2015

Making the Best of Limited Memory in Multi-Player Discounted Sum Games

In this paper, we establish the existence of optimal bounded memory strategy profiles in multi-player discounted sum games. We introduce a non-deterministic approach to compute optimal strategy profiles with bounded memory. Our approach can be used to obtain optimal rewards in a setting where a powerful player selects the strategies of all players for Nash and leader equilibria, where in leader equilibria the Nash condition is waived for the strategy of this powerful player. The resulting strategy profiles are optimal for this player among all strategy profiles that respect the given memory bound, and the related decision problem is NP-complete. We also provide simple examples, which show that having more memory will improve the optimal strategy profile, and that sufficient memory to obtain optimal strategy profiles cannot be inferred from the structure of the game.

cs.GT↗