Time and Parallelizability Results for Parity Games with Bounded Treewidth

@inproceedings{Fearnley2012TimeAP,
  title={Time and Parallelizability Results for Parity Games with Bounded Treewidth},
  author={John Fearnley and Sven Schewe},
  booktitle={ICALP},
  year={2012}
}
Parity games are a much researched class of games in NP ∩ 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 show that, if a tree decomposition is provided, then parity games with bounded treewidth can be solved in O(k3k+2 ·n2 ·(d+1)3k) time, where n, k, and d are the size, treewidth, and number of priorities in the parity game. This significantly improves over previously… 

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References

SHOWING 1-10 OF 23 REFERENCES

Parity Games on Graphs with Medium Tree-Width

TLDR
This is the fastest known algorithm for parity games whose tree-width k satisfies (in standard asymptotic notation) k ∈ ω(log n) and k ∉ o(√n/ log n).

Clique-Width and Parity Games

TLDR
This work presents a polynomial-time algorithm for parity games on graphs of bounded clique-width (class of graphs containing e.g. complete bipartite graphs and cliques), thus completing the picture of the exact complexity of solving parity games.

DAG-Width and Parity Games

TLDR
The natural adaptation of the cops-and-robber game to directed graphs is considered and it is shown that monotone strategies in the game yield a measure with an associated notion of graph decomposition that can be seen to describe how close a directed graph is to a directed acyclic graph (DAG).

The dag-width of directed graphs

A deterministic subexponential algorithm for solving parity games

TLDR
This work uses a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games, and is almost as fast as the randomized algorithms mentioned above.

Fast Mu-Calculus Model Checking when Tree-Width Is Bounded

We show that the model checking problem for μ-calculus on graphs of bounded tree-width can be solved in time linear in the size of the system. The result is presented by first showing a related

A linear time algorithm for finding tree-decompositions of small treewidth

TLDR
Every minor-closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm that determines whether the treewidth of G is at most k, and if so, finds a treedecomposition of G withtreewidth at mostK.

All Structured Programs have Small Tree-Width and Good Register Allocation

TLDR
The register allocation problem for an imperative program is often modelled as the coloring problem of the interference graph of the control-flowGraph of the program, which cannot in general color within a factor O(n ɛ ) from optimality unless NP=P.

Infinite Games Played on Finite Graphs

Efficient Approximation for Triangulation of Minimum Treewidth

TLDR
Four novel approximation algorithms for finding triangulations of minimum treewidth for graphs are presented, improving on the running times of algorithms by Robertson and Seymour, and Becker and Geiger that approximate the optimum by factors of 4 and 32/3.