176 research outputs found
Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space
A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems.
In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal
Local Strategy Improvement for Parity Game Solving
The problem of solving a parity game is at the core of many problems in model
checking, satisfiability checking and program synthesis. Some of the best
algorithms for solving parity game are strategy improvement algorithms. These
are global in nature since they require the entire parity game to be present at
the beginning. This is a distinct disadvantage because in many applications one
only needs to know which winning region a particular node belongs to, and a
witnessing winning strategy may cover only a fractional part of the entire game
graph.
We present a local strategy improvement algorithm which explores the game
graph on-the-fly whilst performing the improvement steps. We also compare it
empirically with existing global strategy improvement algorithms and the
currently only other local algorithm for solving parity games. It turns out
that local strategy improvement can outperform these others by several orders
of magnitude
Three notes on the complexity of model checking fixpoint logic with chop
This paper analyses the complexity of model checking fixpoint logic with Chop – an extension of the
modal μ-calculus with a sequential composition operator. It uses two known game-based characterisations
to derive the following results: the combined model checking complexity as well as the data complexity
of FLC are EXPTIME-complete. This is already the case for its alternation-free fragment. The expression
complexity of FLC is trivially P-hard and limited from above by the complexity of solving a
parity game, i.e. in UP ∩ co-UP. For any fragment of fixed alternation depth, in particular alternation-
free formulas it is P-complete
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