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39
Concurrent Reachability Games
, 2008
"... We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objecti ..."
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Cited by 49 (20 self)
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We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zerosum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type1 states, player 1 has a deterministic strategy to always reach the target. From type2 states, player 1 has a randomized strategy to reach the target with probability 1. From type3 states, player 1 has for every real ε> 0 a randomized strategy to reach the target with probability greater than 1 − ε. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type1 states in linear time, and type2 and type3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.
From nondeterministic Büchi and Streett automata to deterministic parity automata
 In 21st Symposium on Logic in Computer Science (LICS’06
, 2006
"... Determinization and complementation are fundamental notions in computer science. When considering finite automata on finite words determinization gives also a solution to complementation. Given a nondeterministic finite automaton there exists an exponential construction that gives a deterministic au ..."
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Cited by 45 (3 self)
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Determinization and complementation are fundamental notions in computer science. When considering finite automata on finite words determinization gives also a solution to complementation. Given a nondeterministic finite automaton there exists an exponential construction that gives a deterministic automaton for the same language. Dualizing the set of accepting states gives an automaton for the complement language. In the theory of automata on infinite words, determinization and complementation are much more involved. Safra provides determinization constructions for Büchi and Streett automata that result in deterministic Rabin automata. For a Büchi automaton with n states, Safra constructs a deterministic Rabin automaton with n O(n) states and n pairs. For a Streett automaton with n states and k pairs, Safra constructs a deterministic Rabin automaton with (nk) O(nk) states and n(k + 1) pairs. Here, we reconsider Safra’s determinization constructions. We show how to construct automata with fewer states and, most importantly, parity acceptance condition. Specifically, starting from a nondeterministic Büchi automaton with n states our construction yields a deterministic parity automaton with n 2n+2 states and index 2n (instead of a Rabin automaton with (12) n n 2n states and n pairs). Starting from a nondeterministic Streett automaton with n states and k pairs our construction yields a deterministic parity automaton with n n(k+2)+2 (k+1) 2n(k+1) states and index 2n(k + 1) (instead of a Rabin automaton with (12) n(k+1) n n(k+2) (k+1) 2n(k+1) states and n(k+1) pairs). The parity condition is much simpler than the Rabin condition. In applications such as solving games and emptiness of tree automata handling the Rabin condition involves an additional multiplier of n 2 n! (or (n(k + 1)) 2 (n(k + 1))! in the case of Streett) which is saved using our construction.
Subexponential Parameterized Algorithms on Graphs of Bounded Genus and HMinorFree Graphs
, 2003
"... We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum m ..."
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Cited by 41 (13 self)
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We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, cliquetransversal set, and many others restricted to bounded genus graphs. Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes as special cases all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, singlecrossingminorfree graphs, and/or map graphs; we extend these results to apply to boundedgenus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a boundedgenus graph that excludes some planar graph H as a minor. This bound depends linearly on the size (H) of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graphminors work of Robertson and Seymour. Building on these results...
An exponential lower bound for the parity game strategy improvement algorithm as we know it
 In Proc. of 24th LICS
, 2009
"... This paper presents a new lower bound for the discrete strategy improvement algorithm for solving parity games due to Vöge and Jurdziński. First, we informally show which structures are difficult to solve for the algorithm. Second, we outline a family of games on which the algorithm requires exponen ..."
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Cited by 22 (9 self)
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This paper presents a new lower bound for the discrete strategy improvement algorithm for solving parity games due to Vöge and Jurdziński. First, we informally show which structures are difficult to solve for the algorithm. Second, we outline a family of games on which the algorithm requires exponentially many strategy iterations, answering in the negative the longstanding question whether this algorithm runs in polynomial time. Additionally we note that the same family of games can be used to prove a similar result w.r.t. the strategy improvement variant by Schewe as well as the strategy iteration for solving discounted payoff games due to Puri. 1.
An optimal strategy improvement algorithm for solving parity games
, 2007
"... Abstract. This paper presents a novel strategy improvement algorithm for parity and payoff games, which is guaranteed to select, in each improvement step, an optimal combination of local strategy modifications. Current strategy improvement methods stepwise improve the strategy of one player with res ..."
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Cited by 20 (0 self)
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Abstract. This paper presents a novel strategy improvement algorithm for parity and payoff games, which is guaranteed to select, in each improvement step, an optimal combination of local strategy modifications. Current strategy improvement methods stepwise improve the strategy of one player with respect to some ranking function, using an algorithm with two distinct phases: They first choose a modification to the strategy of one player from a list of locally profitable changes, and subsequently evaluate the modified strategy. This separation is unfortunate, because current strategy improvement algorithms have no effective means to predict the global effect of the individual local modifications beyond classifying them as profitable, adversarial, or stale. Furthermore, they are completely blind towards the cross effect of different modifications: Applying one profitable modification may render all other profitable modifications adversarial. Our new construction overcomes the traditional separation between choosing and evaluating the modification to the strategy. It thus improves over current strategy improvement algorithms by providing the optimal improvement in every step, selecting the best combination of local updates from a superset of all profitable and stale changes. 1
On the Power of Imperfect Information
, 2008
"... We present a polynomialtime reduction from parity games with imperfect information to safety games with imperfect information. Similar reductions for games with perfect information typically increase the game size exponentially. Our construction avoids such a blowup by using imperfectinformationto ..."
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Cited by 13 (5 self)
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We present a polynomialtime reduction from parity games with imperfect information to safety games with imperfect information. Similar reductions for games with perfect information typically increase the game size exponentially. Our construction avoids such a blowup by using imperfectinformationtorealisesuccinctcounterswhich cover a range exponentially larger than their size. In particular, the reduction shows that the problem of solving imperfectinformation games with safety conditions is EXPTIMEcomplete.
Solving Parity Games in Practice
"... Abstract. Parity games are 2player games of perfect information and infinite duration that have important applications in automata theory and decision procedures (validity as well as model checking) for temporal logics. In this paper we investigate practical aspects of solving parity games. The mai ..."
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Cited by 10 (8 self)
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Abstract. Parity games are 2player games of perfect information and infinite duration that have important applications in automata theory and decision procedures (validity as well as model checking) for temporal logics. In this paper we investigate practical aspects of solving parity games. The main contribution is a suggestion on how to solve parity games efficiently in practice: we present a generic solver that intertwines optimisations with any of the existing parity game algorithms which is only called on parts of a game that cannot be solved faster by simpler methods. This approach is evaluated empirically on a series of benchmarking games from the aforementioned application domains, showing that using this approach vastly speeds up the solving process. As a sideeffect we obtain the surprising observation that Zielonka’s recursive algorithm is the best parity game solver in practice. 1
A deterministic subexponential algorithm for solving parity games
 In SODA
, 2006
"... Abstract. The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex a ..."
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Cited by 9 (0 self)
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Abstract. The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matouˇsek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. The new algorithm, like the existing randomized subexponential algorithms, uses only polynomial space, and it is almost as fast as the randomized subexponential algorithms mentioned above. Key words. analysis of algorithms and problem complexity, specification and verification, 2player games, games on graphs, discretetime games AMS subject classifications. 68Q25, 68Q60, 91A05, 91A43, 91A50. 1. Introduction. A parity game [11, 15] is played on a directed graph (V, E) by two players, Even and Odd, who move a token from vertex to vertex along the edges of the graph so that an infinite path is formed. A partition (V0, V1) is given of the set V of vertices: player Even moves if the token is at a vertex of V0 and player