@inproceedings{6822,
abstract = {In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the qualitative winner or quantitative payoff of the game. In bidding games, in each turn, we hold an auction between the two players to determine which player moves the token. Bidding games have largely been studied with concrete bidding mechanisms that are variants of a first-price auction: in each turn both players simultaneously submit bids, the higher
bidder moves the token, and pays his bid to the lower bidder in Richman bidding, to the bank in poorman bidding, and in taxman bidding, the bid is split between the other player and the bank according to a predefined constant factor. Bidding games are deterministic games. They have an intriguing connection with a fragment of stochastic games called
randomturn games. We study, for the first time, a combination of bidding games with probabilistic behavior; namely, we study bidding games that are played on Markov decision processes, where the players bid for the right to choose the next action, which determines the probability distribution according to which the next vertex is chosen. We study parity and meanpayoff bidding games on MDPs and extend results from the deterministic bidding setting to the probabilistic one.},
author = {Avni, Guy and Henzinger, Thomas A and Ibsen-Jensen, Rasmus and Novotny, Petr},
booktitle = { Proceedings of the 13th International Conference of Reachability Problems},
location = {Brussels, Belgium},
publisher = {Springer},
title = {{Bidding games on Markov decision processes}},
year = {2019},
}
@article{198,
abstract = {We consider a class of students learning a language from a teacher. The situation can be interpreted as a group of child learners receiving input from the linguistic environment. The teacher provides sample sentences. The students try to learn the grammar from the teacher. In addition to just listening to the teacher, the students can also communicate with each other. The students hold hypotheses about the grammar and change them if they receive counter evidence. The process stops when all students have converged to the correct grammar. We study how the time to convergence depends on the structure of the classroom by introducing and evaluating various complexity measures. We find that structured communication between students, although potentially introducing confusion, can greatly reduce some of the complexity measures. Our theory can also be interpreted as applying to the scientific process, where nature is the teacher and the scientists are the students.},
author = {Ibsen-Jensen, Rasmus and Tkadlec, Josef and Chatterjee, Krishnendu and Nowak, Martin},
journal = {Journal of the Royal Society Interface},
number = {140},
publisher = {Royal Society},
title = {{Language acquisition with communication between learners}},
doi = {10.1098/rsif.2018.0073},
volume = {15},
year = {2018},
}
@inproceedings{5788,
abstract = {In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the winner or payoff of the game. Such games are central in formal verification since they model the interaction between a non-terminating system and its environment. We study bidding games in which the players bid for the right to move the token. Two bidding rules have been defined. In Richman bidding, in each round, the players simultaneously submit bids, and the higher bidder moves the token and pays the other player. Poorman bidding is similar except that the winner of the bidding pays the “bank” rather than the other player. While poorman reachability games have been studied before, we present, for the first time, results on infinite-duration poorman games. A central quantity in these games is the ratio between the two players’ initial budgets. The questions we study concern a necessary and sufficient ratio with which a player can achieve a goal. For reachability objectives, such threshold ratios are known to exist for both bidding rules. We show that the properties of poorman reachability games extend to complex qualitative objectives such as parity, similarly to the Richman case. Our most interesting results concern quantitative poorman games, namely poorman mean-payoff games, where we construct optimal strategies depending on the initial ratio, by showing a connection with random-turn based games. The connection in itself is interesting, because it does not hold for reachability poorman games. We also solve the complexity problems that arise in poorman bidding games.},
author = {Avni, Guy and Henzinger, Thomas A and Ibsen-Jensen, Rasmus},
isbn = {9783030046118},
issn = {03029743},
location = {Oxford, UK},
pages = {21--36},
publisher = {Springer},
title = {{Infinite-duration poorman-bidding games}},
doi = {10.1007/978-3-030-04612-5_2},
volume = {11316},
year = {2018},
}
@inproceedings{66,
abstract = {Crypto-currencies are digital assets designed to work as a medium of exchange, e.g., Bitcoin, but they are susceptible to attacks (dishonest behavior of participants). A framework for the analysis of attacks in crypto-currencies requires (a) modeling of game-theoretic aspects to analyze incentives for deviation from honest behavior; (b) concurrent interactions between participants; and (c) analysis of long-term monetary gains. Traditional game-theoretic approaches for the analysis of security protocols consider either qualitative temporal properties such as safety and termination, or the very special class of one-shot (stateless) games. However, to analyze general attacks on protocols for crypto-currencies, both stateful analysis and quantitative objectives are necessary. In this work our main contributions are as follows: (a) we show how a class of concurrent mean-payo games, namely ergodic games, can model various attacks that arise naturally in crypto-currencies; (b) we present the first practical implementation of algorithms for ergodic games that scales to model realistic problems for crypto-currencies; and (c) we present experimental results showing that our framework can handle games with thousands of states and millions of transitions.},
author = {Chatterjee, Krishnendu and Goharshady, Amir and Ibsen-Jensen, Rasmus and Velner, Yaron},
isbn = {978-3-95977-087-3},
location = {Beijing, China},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Ergodic mean-payoff games for the analysis of attacks in crypto-currencies.}},
doi = {10.4230/LIPIcs.CONCUR.2018.11},
volume = {118},
year = {2018},
}
@inproceedings{5967,
abstract = {The Big Match is a multi-stage two-player game. In each stage Player 1 hides one or two pebbles in his hand, and his opponent has to guess that number; Player 1 loses a point if Player 2 is correct, and otherwise he wins a point. As soon as Player 1 hides one pebble, the players cannot change their choices in any future stage.
Blackwell and Ferguson (1968) give an ε-optimal strategy for Player 1 that hides, in each stage, one pebble with a probability that depends on the entire past history. Any strategy that depends just on the clock or on a finite memory is worthless. The long-standing natural open problem has been whether every strategy that depends just on the clock and a finite memory is worthless. We prove that there is such a strategy that is ε-optimal. In fact, we show that just two states of memory are sufficient.
},
author = {Hansen, Kristoffer Arnsfelt and Ibsen-Jensen, Rasmus and Neyman, Abraham},
booktitle = {Proceedings of the 2018 ACM Conference on Economics and Computation - EC '18},
isbn = {9781450358293},
location = {Ithaca, NY, United States},
pages = {149--150},
publisher = {ACM Press},
title = {{The Big Match with a clock and a bit of memory}},
doi = {10.1145/3219166.3219198},
year = {2018},
}
@article{6009,
abstract = {We study algorithmic questions wrt algebraic path properties in concurrent systems, where the transitions of the system are labeled from a complete, closed semiring. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural problems that arise in program analysis. We consider that each component of the concurrent system is a graph with constant treewidth, a property satisfied by the controlflow graphs of most programs. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis. The study of multiple queries allows us to consider the tradeoff between the resource usage of the one-time preprocessing and for each individual query. The traditional approach constructs the product graph of all components and applies the best-known graph algorithm on the product. In this approach, even the answer to a single query requires the transitive closure (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time.
Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time, each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results showing that the worst-case running time of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (i.e., improving the worst-case bound for the shortest path problem in general graphs). Preliminary experimental results show that our algorithms perform favorably on several benchmarks.
},
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Goharshady, Amir Kafshdar and Pavlogiannis, Andreas},
issn = {0164-0925},
journal = {ACM Transactions on Programming Languages and Systems},
number = {3},
publisher = {Association for Computing Machinery (ACM)},
title = {{Algorithms for algebraic path properties in concurrent systems of constant treewidth components}},
doi = {10.1145/3210257},
volume = {40},
year = {2018},
}
@article{465,
abstract = {The edit distance between two words w 1 , w 2 is the minimal number of word operations (letter insertions, deletions, and substitutions) necessary to transform w 1 to w 2 . The edit distance generalizes to languages L 1 , L 2 , where the edit distance from L 1 to L 2 is the minimal number k such that for every word from L 1 there exists a word in L 2 with edit distance at most k . We study the edit distance computation problem between pushdown automata and their subclasses. The problem of computing edit distance to a pushdown automaton is undecidable, and in practice, the interesting question is to compute the edit distance from a pushdown automaton (the implementation, a standard model for programs with recursion) to a regular language (the specification). In this work, we present a complete picture of decidability and complexity for the following problems: (1) deciding whether, for a given threshold k , the edit distance from a pushdown automaton to a finite automaton is at most k , and (2) deciding whether the edit distance from a pushdown automaton to a finite automaton is finite. },
author = {Chatterjee, Krishnendu and Henzinger, Thomas A and Ibsen-Jensen, Rasmus and Otop, Jan},
issn = {18605974},
journal = {Logical Methods in Computer Science},
number = {3},
publisher = {International Federation of Computational Logic},
title = {{Edit distance for pushdown automata}},
doi = {10.23638/LMCS-13(3:23)2017},
volume = {13},
year = {2017},
}
@inproceedings{551,
abstract = {Evolutionary graph theory studies the evolutionary dynamics in a population structure given as a connected graph. Each node of the graph represents an individual of the population, and edges determine how offspring are placed. We consider the classical birth-death Moran process where there are two types of individuals, namely, the residents with fitness 1 and mutants with fitness r. The fitness indicates the reproductive strength. The evolutionary dynamics happens as follows: in the initial step, in a population of all resident individuals a mutant is introduced, and then at each step, an individual is chosen proportional to the fitness of its type to reproduce, and the offspring replaces a neighbor uniformly at random. The process stops when all individuals are either residents or mutants. The probability that all individuals in the end are mutants is called the fixation probability, which is a key factor in the rate of evolution. We consider the problem of approximating the fixation probability. The class of algorithms that is extremely relevant for approximation of the fixation probabilities is the Monte-Carlo simulation of the process. Previous results present a polynomial-time Monte-Carlo algorithm for undirected graphs when r is given in unary. First, we present a simple modification: instead of simulating each step, we discard ineffective steps, where no node changes type (i.e., either residents replace residents, or mutants replace mutants). Using the above simple modification and our result that the number of effective steps is concentrated around the expected number of effective steps, we present faster polynomial-time Monte-Carlo algorithms for undirected graphs. Our algorithms are always at least a factor O(n2/ log n) faster as compared to the previous algorithms, where n is the number of nodes, and is polynomial even if r is given in binary. We also present lower bounds showing that the upper bound on the expected number of effective steps we present is asymptotically tight for undirected graphs. },
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Nowak, Martin},
booktitle = {Leibniz International Proceedings in Informatics},
isbn = {978-395977046-0},
location = {Aalborg, Denmark},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Faster Monte Carlo algorithms for fixation probability of the Moran process on undirected graphs}},
doi = {10.4230/LIPIcs.MFCS.2017.61},
volume = {83},
year = {2017},
}
@inproceedings{553,
abstract = {We consider two player, zero-sum, finite-state concurrent reachability games, played for an infinite number of rounds, where in every round, each player simultaneously and independently of the other players chooses an action, whereafter the successor state is determined by a probability distribution given by the current state and the chosen actions. Player 1 wins iff a designated goal state is eventually visited. We are interested in the complexity of stationary strategies measured by their patience, which is defined as the inverse of the smallest non-zero probability employed. Our main results are as follows: We show that: (i) the optimal bound on the patience of optimal and -optimal strategies, for both players is doubly exponential; and (ii) even in games with a single non-absorbing state exponential (in the number of actions) patience is necessary. },
author = {Chatterjee, Krishnendu and Hansen, Kristofer and Ibsen-Jensen, Rasmus},
booktitle = {Leibniz International Proceedings in Informatics},
isbn = {978-395977046-0},
location = {Aalborg, Denmark},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Strategy complexity of concurrent safety games}},
doi = {10.4230/LIPIcs.MFCS.2017.55},
volume = {83},
year = {2017},
}
@inproceedings{1340,
abstract = {We study repeated games with absorbing states, a type of two-player, zero-sum concurrent mean-payoff games with the prototypical example being the Big Match of Gillete (1957). These games may not allow optimal strategies but they always have ε-optimal strategies. In this paper we design ε-optimal strategies for Player 1 in these games that use only O(log log T) space. Furthermore, we construct strategies for Player 1 that use space s(T), for an arbitrary small unbounded non-decreasing function s, and which guarantee an ε-optimal value for Player 1 in the limit superior sense. The previously known strategies use space Ω(log T) and it was known that no strategy can use constant space if it is ε-optimal even in the limit superior sense. We also give a complementary lower bound. Furthermore, we also show that no Markov strategy, even extended with finite memory, can ensure value greater than 0 in the Big Match, answering a question posed by Neyman [11].},
author = {Hansen, Kristoffer and Ibsen-Jensen, Rasmus and Koucký, Michal},
location = {Liverpool, United Kingdom},
pages = {64 -- 76},
publisher = {Springer},
title = {{The big match in small space}},
doi = {10.1007/978-3-662-53354-3_6},
volume = {9928},
year = {2016},
}
@inproceedings{478,
abstract = {Magic: the Gathering is a game about magical combat for any number of players. Formally it is a zero-sum, imperfect information stochastic game that consists of a potentially unbounded number of steps. We consider the problem of deciding if a move is legal in a given single step of Magic. We show that the problem is (a) coNP-complete in general; and (b) in P if either of two small sets of cards are not used. Our lower bound holds even for single-player Magic games. The significant aspects of our results are as follows: First, in most real-life game problems, the task of deciding whether a given move is legal in a single step is trivial, and the computationally hard task is to find the best sequence of legal moves in the presence of multiple players. In contrast, quite uniquely our hardness result holds for single step and with only one-player. Second, we establish efficient algorithms for important special cases of Magic.},
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus},
location = {The Hague, Netherlands},
pages = {1432 -- 1439},
publisher = {IOS Press},
title = {{The complexity of deciding legality of a single step of magic: The gathering}},
doi = {10.3233/978-1-61499-672-9-1432},
volume = {285},
year = {2016},
}
@inproceedings{1182,
abstract = {Balanced knockout tournaments are ubiquitous in sports competitions and are also used in decisionmaking and elections. The traditional computational question, that asks to compute a draw (optimal draw) that maximizes the winning probability for a distinguished player, has received a lot of attention. Previous works consider the problem where the pairwise winning probabilities are known precisely, while we study how robust is the winning probability with respect to small errors in the pairwise winning probabilities. First, we present several illuminating examples to establish: (a) there exist deterministic tournaments (where the pairwise winning probabilities are 0 or 1) where one optimal draw is much more robust than the other; and (b) in general, there exist tournaments with slightly suboptimal draws that are more robust than all the optimal draws. The above examples motivate the study of the computational problem of robust draws that guarantee a specified winning probability. Second, we present a polynomial-time algorithm for approximating the robustness of a draw for sufficiently small errors in pairwise winning probabilities, and obtain that the stated computational problem is NP-complete. We also show that two natural cases of deterministic tournaments where the optimal draw could be computed in polynomial time also admit polynomial-time algorithms to compute robust optimal draws.},
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Tkadlec, Josef},
location = {New York, NY, USA},
pages = {172 -- 179},
publisher = {AAAI Press},
title = {{Robust draws in balanced knockout tournaments}},
volume = {2016-January},
year = {2016},
}
@inproceedings{1071,
abstract = {We consider data-structures for answering reachability and distance queries on constant-treewidth graphs with n nodes, on the standard RAM computational model with wordsize W=Theta(log n). Our first contribution is a data-structure that after O(n) preprocessing time, allows (1) pair reachability queries in O(1) time; and (2) single-source reachability queries in O(n/log n) time. This is (asymptotically) optimal and is faster than DFS/BFS when answering more than a constant number of single-source queries. The data-structure uses at all times O(n) space. Our second contribution is a space-time tradeoff data-structure for distance queries. For any epsilon in [1/2,1], we provide a data-structure with polynomial preprocessing time that allows pair queries in O(n^{1-\epsilon} alpha(n)) time, where alpha is the inverse of the Ackermann function, and at all times uses O(n^epsilon) space. The input graph G is not considered in the space complexity. },
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Pavlogiannis, Andreas},
location = {Aarhus, Denmark},
publisher = {Schloss Dagstuhl- Leibniz-Zentrum fur Informatik},
title = {{Optimal reachability and a space time tradeoff for distance queries in constant treewidth graphs}},
doi = {10.4230/LIPIcs.ESA.2016.28},
volume = {57},
year = {2016},
}
@inproceedings{1437,
abstract = {We study algorithmic questions for concurrent systems where the transitions are labeled from a complete, closed semiring, and path properties are algebraic with semiring operations. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural problems that arise in program analysis. We consider that each component of the concurrent system is a graph with constant treewidth, a property satisfied by the controlflow graphs of most programs. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis. The study of multiple queries allows us to consider the tradeoff between the resource usage of the one-time preprocessing and for each individual query. The traditional approach constructs the product graph of all components and applies the best-known graph algorithm on the product. In this approach, even the answer to a single query requires the transitive closure (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time. Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time, each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results showing that the worst-case running time of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (i.e., improving the worst-case bound for the shortest path problem in general graphs). Preliminary experimental results show that our algorithms perform favorably on several benchmarks.},
author = {Chatterjee, Krishnendu and Goharshady, Amir and Ibsen-Jensen, Rasmus and Pavlogiannis, Andreas},
location = {St. Petersburg, FL, USA},
pages = {733 -- 747},
publisher = {ACM},
title = {{Algorithms for algebraic path properties in concurrent systems of constant treewidth components}},
doi = {10.1145/2837614.2837624},
volume = {20-22},
year = {2016},
}
@misc{5430,
abstract = {We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean- payoff property, the ratio property, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with constant treewidth, and it is well-known that the control-flow graphs of most programs have constant treewidth. Let n denote the number of nodes of a graph, m the number of edges (for constant treewidth graphs m = O ( n ) ) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for constant treewidth graphs we present an algorithm that approximates the mean-payoff value within a mul- tiplicative factor of ∊ in time O ( n · log( n/∊ )) and linear space, as compared to the classical algorithms that require quadratic time. Second, for the ratio property we present an algorithm that for constant treewidth graphs works in time O ( n · log( | a · b · n | )) = O ( n · log( n · W )) , when the output is a b , as compared to the previously best known algorithm with running time O ( n 2 · log( n · W )) . Third, for the minimum initial credit problem we show that (i) for general graphs the problem can be solved in O ( n 2 · m ) time and the associated decision problem can be solved in O ( n · m ) time, improving the previous known O ( n 3 · m · log( n · W )) and O ( n 2 · m ) bounds, respectively; and (ii) for constant treewidth graphs we present an algorithm that requires O ( n · log n ) time, improving the previous known O ( n 4 · log( n · W )) bound. We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks.},
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Pavlogiannis, Andreas},
issn = {2664-1690},
pages = {31},
publisher = {IST Austria},
title = {{Faster algorithms for quantitative verification in constant treewidth graphs}},
doi = {10.15479/AT:IST-2015-319-v1-1},
year = {2015},
}
@article{1559,
abstract = {There are deep, yet largely unexplored, connections between computer science and biology. Both disciplines examine how information proliferates in time and space. Central results in computer science describe the complexity of algorithms that solve certain classes of problems. An algorithm is deemed efficient if it can solve a problem in polynomial time, which means the running time of the algorithm is a polynomial function of the length of the input. There are classes of harder problems for which the fastest possible algorithm requires exponential time. Another criterion is the space requirement of the algorithm. There is a crucial distinction between algorithms that can find a solution, verify a solution, or list several distinct solutions in given time and space. The complexity hierarchy that is generated in this way is the foundation of theoretical computer science. Precise complexity results can be notoriously difficult. The famous question whether polynomial time equals nondeterministic polynomial time (i.e., P = NP) is one of the hardest open problems in computer science and all of mathematics. Here, we consider simple processes of ecological and evolutionary spatial dynamics. The basic question is: What is the probability that a new invader (or a new mutant)will take over a resident population?We derive precise complexity results for a variety of scenarios. We therefore show that some fundamental questions in this area cannot be answered by simple equations (assuming that P is not equal to NP).},
author = {Ibsen-Jensen, Rasmus and Chatterjee, Krishnendu and Nowak, Martin},
journal = {PNAS},
number = {51},
pages = {15636 -- 15641},
publisher = {National Academy of Sciences},
title = {{Computational complexity of ecological and evolutionary spatial dynamics}},
doi = {10.1073/pnas.1511366112},
volume = {112},
year = {2015},
}
@article{524,
abstract = {We consider concurrent games played by two players on a finite-state graph, where in every round the players simultaneously choose a move, and the current state along with the joint moves determine the successor state. We study the most fundamental objective for concurrent games, namely, mean-payoff or limit-average objective, where a reward is associated to each transition, and the goal of player 1 is to maximize the long-run average of the rewards, and the objective of player 2 is strictly the opposite (i.e., the games are zero-sum). The path constraint for player 1 could be qualitative, i.e., the mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative, i.e., a given threshold between the minimal and maximal reward. We consider the computation of the almost-sure (resp. positive) winning sets, where player 1 can ensure that the path constraint is satisfied with probability 1 (resp. positive probability). Almost-sure winning with qualitative constraint exactly corresponds to the question of whether there exists a strategy to ensure that the payoff is the maximal reward of the game. Our main results for qualitative path constraints are as follows: (1) we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost-sure (resp. positive) winning against all player-2 strategies, or player 2 has a spoiling strategy to falsify almost-sure (resp. positive) winning against all player-1 strategies; (2) we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost-sure and positive winning for both players; and (3) we present quadratic time algorithms to compute the almost-sure and the positive winning sets, matching the best known bound of the algorithms for much simpler problems (such as reachability objectives). For quantitative constraints we show that a polynomial time solution for the almost-sure or the positive winning set would imply a solution to a long-standing open problem (of solving the value problem of turn-based deterministic mean-payoff games) that is not known to be solvable in polynomial time.},
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus},
journal = {Information and Computation},
number = {6},
pages = {2 -- 24},
publisher = {Elsevier},
title = {{Qualitative analysis of concurrent mean payoff games}},
doi = {10.1016/j.ic.2015.03.009},
volume = {242},
year = {2015},
}
@misc{5431,
abstract = {We consider finite-state concurrent stochastic games, played by k>=2 players for an infinite number of rounds, where in every round, each player simultaneously and independently of the other players chooses an action, whereafter the successor state is determined by a probability distribution given by the current state and the chosen actions. We consider reachability objectives that given a target set of states require that some state in the target set is visited, and the dual safety objectives that given a target set require that only states in the target set are visited. We are interested in the complexity of stationary strategies measured by their patience, which is defined as the inverse of the smallest non-zero probability employed.
Our main results are as follows: We show that in two-player zero-sum concurrent stochastic games (with reachability objective for one player and the complementary safety objective for the other player): (i) the optimal bound on the patience of optimal and epsilon-optimal strategies, for both players is doubly exponential; and (ii) even in games with a single non-absorbing state exponential (in the number of actions) patience is necessary. In general we study the class of non-zero-sum games admitting epsilon-Nash equilibria. We show that if there is at least one player with reachability objective, then doubly-exponential patience is needed in general for epsilon-Nash equilibrium strategies, whereas in contrast if all players have safety objectives, then the optimal bound on patience for epsilon-Nash equilibrium strategies is only exponential.},
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Hansen, Kristoffer},
issn = {2664-1690},
pages = {25},
publisher = {IST Austria},
title = {{The patience of concurrent stochastic games with safety and reachability objectives}},
doi = {10.15479/AT:IST-2015-322-v1-1},
year = {2015},
}
@misc{5437,
abstract = {We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean-payoff property, the ratio property, and the minimum initial credit for energy property.
The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with constant treewidth, and it is well-known that the control-flow graphs of most programs have constant treewidth. Let $n$ denote the number of nodes of a graph, $m$ the number of edges (for constant treewidth graphs $m=O(n)$) and $W$ the largest absolute value of the weights.
Our main theoretical results are as follows.
First, for constant treewidth graphs we present an algorithm that approximates the mean-payoff value within a multiplicative factor of $\epsilon$ in time $O(n \cdot \log (n/\epsilon))$ and linear space, as compared to the classical algorithms that require quadratic time. Second, for the ratio property we present an algorithm that for constant treewidth graphs works in time $O(n \cdot \log (|a\cdot b|))=O(n\cdot\log (n\cdot W))$, when the output is $\frac{a}{b}$, as compared to the previously best known algorithm with running time $O(n^2 \cdot \log (n\cdot W))$. Third, for the minimum initial credit problem we show that (i)~for general graphs the problem can be solved in $O(n^2\cdot m)$ time and the associated decision problem can be solved in $O(n\cdot m)$ time, improving the previous known $O(n^3\cdot m\cdot \log (n\cdot W))$ and $O(n^2 \cdot m)$ bounds, respectively; and (ii)~for constant treewidth graphs we present an algorithm that requires $O(n\cdot \log n)$ time, improving the previous known $O(n^4 \cdot \log (n \cdot W))$ bound.
We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks. },
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Pavlogiannis, Andreas},
issn = {2664-1690},
pages = {27},
publisher = {IST Austria},
title = {{Faster algorithms for quantitative verification in constant treewidth graphs}},
doi = {10.15479/AT:IST-2015-330-v2-1},
year = {2015},
}
@misc{5432,
abstract = {Evolution occurs in populations of reproducing individuals. The structure of the population affects the outcome of the evolutionary process. Evolutionary graph theory is a powerful approach to study this phenomenon. There are two graphs. The interaction graph specifies who interacts with whom in the context of evolution.The replacement graph specifies who competes with whom for reproduction.
The vertices of the two graphs are the same, and each vertex corresponds to an individual of the population. A key quantity is the fixation probability of a new mutant. It is defined as the probability that a newly introduced mutant (on a single vertex) generates a lineage of offspring which eventually takes over the entire population of resident individuals. The basic computational questions are as follows: (i) the qualitative question asks whether the fixation probability is positive; and (ii) the quantitative approximation question asks for an approximation of the fixation probability.
Our main results are:
(1) We show that the qualitative question is NP-complete and the quantitative approximation question is #P-hard in the special case when the interaction and the replacement graphs coincide and even with the restriction that the resident individuals do not reproduce (which corresponds to an invading population taking over an empty structure).
(2) We show that in general the qualitative question is PSPACE-complete and the quantitative approximation question is PSPACE-hard and can be solved in exponential time.
},
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Nowak, Martin},
issn = {2664-1690},
pages = {29},
publisher = {IST Austria},
title = {{The complexity of evolutionary games on graphs}},
doi = {10.15479/AT:IST-2015-323-v1-1},
year = {2015},
}