# News for January 2019

Minimax Testing of Identity to a Reference Ergodic Markov Chain, by Geoffrey Wolfer and Aryeh Kontorovich (arXiv). This work studies distributional identity testing on Markov chains from a single trajectory, as recently introduced by Daskalakis, Dikkala, and Gravin: we wish to test whether a Markov chain is equal to some reference chain, or far from it. This improves on previous work by considering a stronger distance measure than before, and showing that the sample complexity only depends on properties of the reference chain (which we are trying to test identity to). It additionally proves instance-by-instance bounds (where the sample complexity depends on properties of the specific chain we wish to test identity to).

Almost Optimal Distribution-free Junta Testing, by Nader H. Bshouty (arXiv). This paper provides a $$\tilde O(k/\varepsilon)$$-query algorithm with two-sided error for testing if a Boolean function is a $$k$$-junta (that is, its value depends only on $$k$$ of its variables) in the distribution-free model (where distance is measured with respect to an unknown distribution from which we can sample). This complexity is a quadratic improvement over the $$\tilde O(k^2)/\varepsilon$$-query algorithm of Chen, Liu, Servedio, Sheng, and Xie. This complexity is also near-optimal, as shown in a lower bound by Saglam (which we covered back in August).

Exponentially Faster Massively Parallel Maximal Matching, by Soheil Behnezhad, MohammadTaghi Hajiaghayi, and David G. Harris (arXiv). The authors consider maximal matching in the Massively Parallel Computation (MPC) model. They show that one can compute a maximal matching in $$O(\log \log \Delta)$$-rounds, with $$O(n)$$ space per machine. This is an exponential improvement over the previous works, which required either $$\Omega(\log n)$$ rounds or $$n^{1 + \Omega(1)}$$ space per machine. Corollaries of their result include approximation algorithms for vertex cover, maximum matching, and weighted maximum matching.