20 research outputs found
Parity Games on Undirected Graphs
International audienceWe examine the complexity of solving parity games in the special case when the underlying game graph is undirected. For strictly alternating games, that is, when the game graph is bipartite between the nodes of the two players, we observe that the solution can be computed in linear time. In contrast, when the assumption of strict alternation is dropped, we show that the problem is as hard in the undirected case as it is in the general, directed, case
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
Finding Optimal Strategies of Almost Acyclic Simple Stochatic Games
The optimal value computation for turned-based stochastic games with
reachability objectives, also known as simple stochastic games, is one of the
few problems in which are not known to be in . However, there
are some cases where these games can be easily solved, as for instance when the
underlying graph is acyclic. In this work, we try to extend this tractability
to several classes of games that can be thought as "almost" acyclic. We give
some fixed-parameter tractable or polynomial algorithms in terms of different
parameters such as the number of cycles or the size of the minimal feedback
vertex set
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
Fast Algorithms for Energy Games in Special Cases
In this paper, we study algorithms for special cases of energy games, a class
of turn-based games on graphs that show up in the quantitative analysis of
reactive systems. In an energy game, the vertices of a weighted directed graph
belong either to Alice or to Bob. A token is moved to a next vertex by the
player controlling its current location, and its energy is changed by the
weight of the edge. Given a fixed starting vertex and initial energy, Alice
wins the game if the energy of the token remains nonnegative at every moment.
If the energy goes below zero at some point, then Bob wins. The problem of
determining the winner in an energy game lies in . It is a long standing open problem whether a polynomial time
algorithm for this problem exists.
We devise new algorithms for three special cases of the problem. The first
two results focus on the single-player version, where either Alice or Bob
controls the whole game graph. We develop an
time algorithm for a game graph controlled by Alice, by providing a reduction
to the All-Pairs Nonnegative Prefix Paths problem (APNP), where is the
maximum weight and is the best exponent for matrix multiplication.
Thus we study the APNP problem separately, for which we develop an
time algorithm. For both problems, we improve
over the state of the art of for small . For the APNP
problem, we also provide a conditional lower bound, which states that there is
no time algorithm for any , unless the APSP
Hypothesis fails. For a game graph controlled by Bob, we obtain a near-linear
time algorithm. Regarding our third result, we present a variant of the value
iteration algorithm, and we prove that it gives an time algorithm for
game graphs without negative cycles