655 research outputs found
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
Graph Searching, Parity Games and Imperfect Information
We investigate the interrelation between graph searching games and games with
imperfect information. As key consequence we obtain that parity games with
bounded imperfect information can be solved in PTIME on graphs of bounded
DAG-width which generalizes several results for parity games on graphs of
bounded complexity. We use a new concept of graph searching where several cops
try to catch multiple robbers instead of just a single robber. The main
technical result is that the number of cops needed to catch r robbers
monotonously is at most r times the DAG-width of the graph. We also explore
aspects of this new concept as a refinement of directed path-width which
accentuates its connection to the concept of imperfect information
The Rabin index of parity games
We study the descriptive complexity of parity games by taking into account
the coloring of their game graphs whilst ignoring their ownership structure.
Colored game graphs are identified if they determine the same winning regions
and strategies, for all ownership structures of nodes. The Rabin index of a
parity game is the minimum of the maximal color taken over all equivalent
coloring functions. We show that deciding whether the Rabin index is at least k
is in PTIME for k=1 but NP-hard for all fixed k > 1. We present an EXPTIME
algorithm that computes the Rabin index by simplifying its input coloring
function. When replacing simple cycle with cycle detection in that algorithm,
its output over-approximates the Rabin index in polynomial time. Experimental
results show that this approximation yields good values in practice.Comment: In Proceedings GandALF 2013, arXiv:1307.416
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
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