43,317 research outputs found
Energy Parity Games
Energy parity games are infinite two-player turn-based games played on
weighted graphs. The objective of the game combines a (qualitative) parity
condition with the (quantitative) requirement that the sum of the weights
(i.e., the level of energy in the game) must remain positive. Beside their own
interest in the design and synthesis of resource-constrained omega-regular
specifications, energy parity games provide one of the simplest model of games
with combined qualitative and quantitative objective. Our main results are as
follows: (a) exponential memory is necessary and sufficient for winning
strategies in energy parity games; (b) the problem of deciding the winner in
energy parity games can be solved in NP \cap coNP; and (c) we give an algorithm
to solve energy parity by reduction to energy games. We also show that the
problem of deciding the winner in energy parity games is polynomially
equivalent to the problem of deciding the winner in mean-payoff parity games,
while optimal strategies may require infinite memory in mean-payoff parity
games. As a consequence we obtain a conceptually simple algorithm to solve
mean-payoff parity games
Positional Determinacy of Games with Infinitely Many Priorities
We study two-player games of infinite duration that are played on finite or
infinite game graphs. A winning strategy for such a game is positional if it
only depends on the current position, and not on the history of the play. A
game is positionally determined if, from each position, one of the two players
has a positional winning strategy.
The theory of such games is well studied for winning conditions that are
defined in terms of a mapping that assigns to each position a priority from a
finite set. Specifically, in Muller games the winner of a play is determined by
the set of those priorities that have been seen infinitely often; an important
special case are parity games where the least (or greatest) priority occurring
infinitely often determines the winner. It is well-known that parity games are
positionally determined whereas Muller games are determined via finite-memory
strategies.
In this paper, we extend this theory to the case of games with infinitely
many priorities. Such games arise in several application areas, for instance in
pushdown games with winning conditions depending on stack contents.
For parity games there are several generalisations to the case of infinitely
many priorities. While max-parity games over omega or min-parity games over
larger ordinals than omega require strategies with infinite memory, we can
prove that min-parity games with priorities in omega are positionally
determined. Indeed, it turns out that the min-parity condition over omega is
the only infinitary Muller condition that guarantees positional determinacy on
all game graphs
Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games
2.5 player parity games combine the challenges posed by 2.5 player
reachability games and the qualitative analysis of parity games. These two
types of problems are best approached with different types of algorithms:
strategy improvement algorithms for 2.5 player reachability games and recursive
algorithms for the qualitative analysis of parity games. We present a method
that - in contrast to existing techniques - tackles both aspects with the best
suited approach and works exclusively on the 2.5 player game itself. The
resulting technique is powerful enough to handle games with several million
states
Time and Parallelizability Results for Parity Games with Bounded Tree and DAG Width
Parity games are a much researched class of games in NP intersect CoNP that
are not known to be in P. Consequently, researchers have considered specialised
algorithms for the case where certain graph parameters are small. In this
paper, we study parity games on graphs with bounded treewidth, and graphs with
bounded DAG width. We show that parity games with bounded DAG width can be
solved in O(n^(k+3) k^(k + 2) (d + 1)^(3k + 2)) time, where n, k, and d are the
size, treewidth, and number of priorities in the parity game. This is an
improvement over the previous best algorithm, given by Berwanger et al., which
runs in n^O(k^2) time. We also show that, if a tree decomposition is provided,
then parity games with bounded treewidth can be solved in O(n k^(k + 5) (d +
1)^(3k + 5)) time. This improves over previous best algorithm, given by
Obdrzalek, which runs in O(n d^(2(k+1)^2)) time. Our techniques can also be
adapted to show that the problem of solving parity games with bounded treewidth
lies in the complexity class NC^2, which is the class of problems that can be
efficiently parallelized. This is in stark contrast to the general parity game
problem, which is known to be P-hard, and thus unlikely to be contained in NC
Benchmarks for Parity Games (extended version)
We propose a benchmark suite for parity games that includes all benchmarks
that have been used in the literature, and make it available online. We give an
overview of the parity games, including a description of how they have been
generated. We also describe structural properties of parity games, and using
these properties we show that our benchmarks are representative. With this work
we provide a starting point for further experimentation with parity games.Comment: The corresponding tool and benchmarks are available from
https://github.com/jkeiren/paritygame-generator. This is an extended version
of the paper that has been accepted for FSEN 201
Solving parity games: Explicit vs symbolic
In this paper we provide a broad investigation of the symbolic approach for solving Parity Games. Specifically, we implement in a fresh tool, called, four symbolic algorithms to solve Parity Games and compare their performances to the corresponding explicit versions for different classes of games. By means of benchmarks, we show that for random games, even for constrained random games, explicit algorithms actually perform better than symbolic algorithms. The situation changes, however, for structured games, where symbolic algorithms seem to have the advantage. This suggests that when evaluating algorithms for parity-game solving, it would be useful to have real benchmarks and not only random benchmarks, as the common practice has been
New Deterministic Algorithms for Solving Parity Games
We study parity games in which one of the two players controls only a small
number of nodes and the other player controls the other nodes of the
game. Our main result is a fixed-parameter algorithm that solves bipartite
parity games in time , and general parity games in
time , where is the number of distinct
priorities and is the number of edges. For all games with 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
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