Courcelle's theorem states that there exists an algorithm that takes as input a graph G of treewidth at most t and a MSO formula ϕ, and determines whether G satisfies ϕ in time f(ϕ,t) · n. It is folklore that the the function f contains a tower of exponentials whose height depends as a linear function of the number of quantifier alternations of the input formula ϕ. A classic reduction of Frick and Grohe shows that, assuming the Exponential Time Hypothesis (ETH), the linear growth of the height of the tower is unavoidable. Nevertheless, there is still a huge gap between existing upper and lower bounds – after all, there is quite a difference between a single exponential and a double exponential running time. In addition, this only gives us a very coarse understanding in the time complexity of Courcelle's theorem. In this paper, we prove a fine-grained version of Courcelle's theorem with nearly ETH-tight dependence on the treewidth parameter t and the quantifier structure of ϕ (specifically, the number of first order and second order variables in each quantifier alternation block).
Motivated by the challenge of testing distributions over high-dimensional or continuous domains, we study distribution testing with respect to bounded classes of distinguishers. A representative task is to use samples from an unknown distribution P over a very large domain to decide between two cases: P = P_𝗋𝖾𝖿 for a fixed reference distribution P_𝗋𝖾𝖿, or there exists a distinguisher f in a bounded class ℱ which witnesses the separation |𝐄_P[f] - 𝐄_P_𝗋𝖾𝖿[f]| > ε. This is the task of identity testing with respect to fooling distance, a name inspired by the conceptual connection with pseudorandomness. (Formally, our model instantiates integral probability metrics from Boolean classes of bounded expressivity.) We show that testing with respect to fooling distance is not only a natural computational problem that admits sample-efficient algorithms even in high-dimensional settings, but also one that reveals and underlies connections between three seemingly unrelated areas of study: testable learning, verification of learning algorithms, and testing of structured distributions (whose "𝒜_k-testing" model our framework extends). These connections yield new results for all of these models, including: 1. Testable proper learners using membership queries for halfspaces and decision trees. 2. A lower bound for testable PAC verification in terms of Rademacher complexity, and a distribution-free verification protocol for disjoint unions of k multidimensional rectangles. 3. Identity testers (with respect to total variation distance) for decision tree distributions and distributions with low-degree polynomial densities, over Boolean and continuous hypercube domains.
In the semi-streaming model, we have an n-vertex graph G=(V,E) whose edges arrive in an arbitrary order in a stream. The goal is to make one or a few passes over the stream, use a limited memory of Õ(n) bits, and output a solution to the problem at hand at the end. A central open question in this area is to determine the best approximation ratio possible for the maximum matching problem via single-pass semi-streaming algorithms. This problem admits a simple 0.5-approximation algorithm, by maintaining a maximal matching greedily, which, despite extensive efforts, has remained the state of the art. Lower bounds for this problem have also been few and far between with best known bounds ruling out better than 1/(1+ln(2)) ∼ 0.590 approximation, using a highly complicated construction motivated by the literature on RS graphs from extremal graph theory. We develop a new framework for proving lower bounds for the semi-streaming matching problem. Our framework abstracts out the extremal graph theory and information theoretic arguments in the lower bounds, and reduces the problem to constructing certain constant-size graphs, which we call blueprints. Not only existing lower bounds can be captured by these blueprints, leading to far simpler and more concise arguments, but also we can design new blueprints that can be used to rule out (8-2√(10))/3 ∼ 0.558-approximation for the semi-streaming matching problem. We believe this approach can be of its own independent interest and lead to further improvements on this tantalizing open question.
