We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu (2007) from planar graphs to bounded-genus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to bounded-genus graphs.
Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomialsized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the gridlike minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter; the only other result with a similar flavor that is known to us is due to Demaine and Hajiaghayi and obtains a doubly-exponential bound on the parameter (albeit, for a more general class of parameterized problems).The second part of our work deals with providing a lower bound to Courcelle's famous theorem from almost two decades ago, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Whereas much work has been done on designing, improving, and applying algorithms on graphs of bounded treewidth, not much is known on the side of lower bounds: what bound on the treewidth of a class of graphs "forbids" polynomial-time parameterized algorithms to decide MSO-sentences? This question has only recently received attention with the first systematic study appearing in [Kreutzer 2009]. Using our results from the first part of our work we can improve on it significantly and establish a strong lower bound for Courcelle's theorem on classes of colored graphs.
Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic (MSO2) can be decided in linear time on any class of graphs of bounded tree-width, or in other words, MSO2 is fixedparameter tractable in linear time on any such class of graphs. From a logical perspective, Courcelle's theorem establishes a sufficient condition, or an upper bound, for tractability of MSO2-model checking.Whereas such upper bounds on the complexity of logics have received significant attention in the literature, almost nothing is known about corresponding lower bounds. In this paper we establish a strong lower bound for the complexity of monadic second-order logic. In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log c n for some small c suffices) then MSO2-model checking is intractable on C (under a suitable assumption from complexity theory).
This paper presents advances in Google's hidden Markov model (HMM)-driven unit selection speech synthesis system. We describe several improvements to the run-time system; these include minimal latency, high-quality and fast refresh cycle for new voices. Traditionally unit selection synthesizers are limited in terms of the amount of data they can handle and the real applications they are built for. That is even more critical for reallife large-scale applications where high-quality is expected and low latency is required given the available computational resources. In this paper we present an optimized engine to handle a large database at runtime, a composite unit search approach for combining diphones and phrase-based units. In addition a new voice building strategy for handling big databases and keeping the building times low is presented.i , o l } refer to its observation vector o l and to a contextual model from
a b s t r a c tWe generalize the linear-time shortest-paths algorithm for planar graphs with nonnegative edge-weights of Henzinger et al. (1994) to work for any proper minor-closed class of graphs. We argue that their algorithm can not be adapted by standard methods to all proper minor-closed classes. By using recent deep results in graph minor theory, we show how to construct an appropriate recursive division in linear time for any graph excluding a fixed minor and how to transform the graph and its division afterwards, so that it has maximum degree three. Based on such a division, the original framework of Henzinger et al. can be applied. Afterwards, we show that using this algorithm, one can implement Mehlhorn's (1988) 2-approximation algorithm for the Steiner tree problem in linear time on these graph classes.
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