10,619 research outputs found
Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games
The McNaughton-Zielonka divide et impera algorithm is the simplest and most
flexible approach available in the literature for determining the winner in a
parity game. Despite its theoretical worst-case complexity and the negative
reputation as a poorly effective algorithm in practice, it has been shown to
rank among the best techniques for the solution of such games. Also, it proved
to be resistant to a lower bound attack, even more than the strategy
improvements approaches, and only recently a family of games on which the
algorithm requires exponential time has been provided by Friedmann. An easy
analysis of this family shows that a simple memoization technique can help the
algorithm solve the family in polynomial time. The same result can also be
achieved by exploiting an approach based on the dominion-decomposition
techniques proposed in the literature. These observations raise the question
whether a suitable combination of dynamic programming and game-decomposition
techniques can improve on the exponential worst case of the original algorithm.
In this paper we answer this question negatively, by providing a robustly
exponential worst case, showing that no intertwining of the above mentioned
techniques can help mitigating the exponential nature of the divide et impera
approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176
Geometric Property (T)
This paper discusses `geometric property (T)'. This is a property of metric
spaces introduced in earlier work of the authors for its applications to
K-theory. Geometric property (T) is a strong form of `expansion property': in
particular for a sequence of finite graphs , it is strictly stronger
than being an expander in the sense that the Cheeger constants
are bounded below.
We show here that geometric property (T) is a coarse invariant, i.e. depends
only on the large-scale geometry of a metric space . We also discuss the
relationships between geometric property (T) and amenability, property (T), and
various coarse geometric notions of a-T-menability. In particular, we show that
property (T) for a residually finite group is characterised by geometric
property (T) for its finite quotients.Comment: Version two corrects some typos and a mistake in the proof of Lemma
8.
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