The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average

We introduce an algorithm which solves mean payoff games in polynomial time on average, assuming the distribution of the games satisfies a flip invariance property on the set of actions associated with every state. The algorithm is a tropical analogue of the shadow-vertex simplex algorithm, which solves mean payoff games via linear feasibility problems over the tropical semiring $(\mathbb{R} \cup \{-\infty\}, \max, +)$. The key ingredient in our approach is that the shadow-vertex pivoting rule can be transferred to tropical polyhedra, and that its computation reduces to optimal assignment problems through Pl\"ucker relations.Comment: 17 pages, 7 figures, appears in 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part

New Deterministic Algorithms for Solving Parity Games

We study parity games in which one of the two players controls only a small number $k$ of nodes and the other player controls the $n-k$ other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time $k^{O(\sqrt{k})}\cdot O(n^3)$, and general parity games in time $(p+k)^{O(\sqrt{k})} \cdot O(pnm)$, where $p$ is the number of distinct priorities and $m$ is the number of edges. For all games with $k = o(n)$ this improves the previously fastest algorithm by Jurdzi{\'n}ski, Paterson, and Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree

Local Strategy Improvement for Parity Game Solving

The problem of solving a parity game is at the core of many problems in model checking, satisfiability checking and program synthesis. Some of the best algorithms for solving parity game are strategy improvement algorithms. These are global in nature since they require the entire parity game to be present at the beginning. This is a distinct disadvantage because in many applications one only needs to know which winning region a particular node belongs to, and a witnessing winning strategy may cover only a fractional part of the entire game graph. We present a local strategy improvement algorithm which explores the game graph on-the-fly whilst performing the improvement steps. We also compare it empirically with existing global strategy improvement algorithms and the currently only other local algorithm for solving parity games. It turns out that local strategy improvement can outperform these others by several orders of magnitude

A parallel algorithm for the enumeration of benzenoid hydrocarbons

We present an improved parallel algorithm for the enumeration of fixed benzenoids B_h containing h hexagonal cells. We can thus extend the enumeration of B_h from the previous best h=35 up to h=50. Analysis of the associated generating function confirms to a very high degree of certainty that $B_h \sim A \kappa^h /h$ and we estimate that the growth constant $\kappa = 5.161930154(8)$ and the amplitude $A=0.2808499(1)$.Comment: 14 pages, 6 figure

Symmetric Strategy Improvement

Symmetry is inherent in the definition of most of the two-player zero-sum games, including parity, mean-payoff, and discounted-payoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where, in each iteration, the strategies of both players are improved simultaneously. We show that symmetric strategy improvement defies Friedmann's traps, which shook the belief in the potential of classic strategy improvement to be polynomial

Oink: an Implementation and Evaluation of Modern Parity Game Solvers

Parity games have important practical applications in formal verification and synthesis, especially to solve the model-checking problem of the modal mu-calculus. They are also interesting from the theory perspective, as they are widely believed to admit a polynomial solution, but so far no such algorithm is known. In recent years, a number of new algorithms and improvements to existing algorithms have been proposed. We implement a new and easy to extend tool Oink, which is a high-performance implementation of modern parity game algorithms. We further present a comprehensive empirical evaluation of modern parity game algorithms and solvers, both on real world benchmarks and randomly generated games. Our experiments show that our new tool Oink outperforms the current state-of-the-art.Comment: Accepted at TACAS 201

Efficient Parallel Strategy Improvement for Parity Games

We study strategy improvement algorithms for solving parity games. While these algorithms are known to solve parity games using a very small number of iterations, experimental studies have found that a high step complexity causes them to perform poorly in practice. In this paper we seek to address this situation. Every iteration of the algorithm must compute a best response, and while the standard way of doing this uses the Bellman-Ford algorithm, we give experimental results that show that one-player strategy improvement significantly outperforms this technique in practice. We then study the best way to implement one-player strategy improvement, and we develop an efficient parallel algorithm for carrying out this task, by reducing the problem to computing prefix sums on a linked list. We report experimental results for these algorithms, and we find that a GPU implementation of this algorithm shows a significant speedup over single-core and multi-core CPU implementations

Radar Doppler velocity of volcanic eruptions: Theoretical considerations and quantitative documentation of changes in eruptive behaviour at Stromboli volcano, Italy

The use of radar Doppler velocimetry for the observation of volcanic activity is new. We used this method to continuously observe the activity of one vent of Stromboli volcano, Italy, from the end of 2000 April until early May. During this period we recorded 702 eruptions, 132 of which occurred before a strong rain storm passed over the island on April 29. In order to interpret the recorded Doppler data we developed a program that simulates different strombolian eruption scenarios, for which we then calculate the theoretical Doppler spectra. Comparing our theoretical data with the observed data we are able to show that most of the eruptions are nearly vertical, although we did observe only one component of the eruption vector with our Doppler radar. One of the most interesting features of the data set is a significant change in eruptive behaviour that correlates with the occurrence of the rain storm: we find that on average the eruption duration increased by a factor of 2, eruptive velocities were much higher and indirect evidence indicates that the average particle diameter of the erupted material decreased. This change may have several causes, but the coincidence with the rain storm may be evidence of magma–water interaction and feedback on the volcanic activity. If the fluid source (rain) changing the eruptive style is at the surface and in near-surface layers then the main control on final eruption dynamics at Stromboli volcano must also be in rather shallow regions

Semiperfect-Information Games

Abstract. Much recent research has focused on the applications of games with!-regular objectives in the control and verication of reactive systems. However, many of the game-based models are ill-suited for these applications, because they assume that each player has complete infor-mation about the state of the system (they are \perfect-information" games). This is because in many situations, a controller does not see the private state of the plant. Such scenarios are naturally modeled by \partial-information " games. On the other hand, these games are in-tractable; for example, partial-information games with simple reachabil-ity objectives are 2EXPTIME-complete. We study the intermediate case of \semiperfect-information " games, where one player has complete knowledge of the state, while the other player has only partial knowledge. This model is appropriate in con-trol situations where a controller must cope with plant behavior that is as adversarial as possible, i.e., the controller has partial informa-tion while the plant has perfect information. As is customary, we as-sume that the controller and plant take turns to make moves. We show that these semiperfect-information turn-based games are equiv-alent to perfect-information concurrent games, where the two play-ers choose their moves simultaneously and independently. Since the perfect-information concurrent games are well-understood, we obtain several results of how semiperfect-information turn-based games dif-fer from perfect-information turn-based games on one hand, and from partial-information turn-based games on the other hand. In particular, semiperfect-information turn-based games can benet from randomized strategies while the perfect-information variety cannot, and semiperfect-information turn-based games are in NP \ coNP for all parity objectives.