1 research outputs found
Parity Games, Imperfect Information and Structural Complexity
We address the problem of solving parity games with imperfect information on
finite graphs of bounded structural complexity. It is a major open problem
whether parity games with perfect information can be solved in PTIME.
Restricting the structural complexity of the game arenas, however, often leads
to efficient algorithms for parity games. Such results are known for graph
classes of bounded tree-width, DAG-width, directed path-width, and
entanglement, which we describe in terms of cops and robber games. Conversely,
the introduction of imperfect information makes the problem more difficult, it
becomes EXPTIME-hard. We analyse the interaction of both approaches.
We use a simple method to measure the amount of "unawareness"' of a player,
the amount of imperfect information. It turns out that if it is unbounded, low
structural complexity does not make the problem simpler. It remains
EXPTIME-hard or PSPACE-hard even on very simple graphs.
For games with bounded imperfect information we analyse the powerset
construction, which is commonly used to convert a game of imperfect information
into an equivalent game with perfect information. This construction preserves
boundedness of directed path-width and DAG-width, but not of entanglement or of
tree-width. Hence, if directed path-width or DAG-width are bounded, parity
games with bounded imperfect information can be solved in PTIME. For DAG-width
we follow two approaches. One leads to a generalization of the known fact that
perfect information parity games are in PTIME if DAG-width is bounded. We prove
this theorem for non-monotone DAG-width. The other approach introduces a cops
and robbers game (with multiple robbers) on directed graphs, considered
Richerby and Thilikos forundirected graphs. We show a tight linear bound for
the number of additional cops needed to capture an additional robber.Comment: arXiv admin note: text overlap with arXiv:1110.557