Faster Isomorphism for pp-Groups of Class 2 and Exponent pp

The group isomorphism problem determines whether two groups, given by their Cayley tables, are isomorphic. For groups with order $n$, an algorithm with $n^{(\log n + O(1))}$ running time, attributed to Tarjan, was proposed in the 1970s [Mil78]. Despite the extensive study over the past decades, the current best group isomorphism algorithm has an $n^{(1 / 4 + o(1))\log n}$ running time [Ros13]. The isomorphism testing for $p$-groups of (nilpotent) class 2 and exponent $p$ has been identified as a major barrier to obtaining an $n^{o(\log n)}$ time algorithm for the group isomorphism problem. Although the $p$-groups of class 2 and exponent $p$ have much simpler algebraic structures than general groups, the best-known isomorphism testing algorithm for this group class also has an $n^{O(\log n)}$ running time. In this paper, we present an isomorphism testing algorithm for $p$-groups of class 2 and exponent $p$ with running time $n^{O((\log n)^{5/6})}$ for any prime $p > 2$. Our result is based on a novel reduction to the skew-symmetric matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop several tools for matrix space analysis, including a matrix space individualization-refinement method and a characterization of the low rank matrix spaces.Comment: Accepted to STOC 202

Learning mixtures of structured distributions over discrete domains

Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n] = \{1,...,n\}.$ We show that if $\mathfrak{C}$ satisfies a rather general condition -- essentially, that each distribution in $\mathfrak{C}$ can be well-approximated by a variable-width histogram with few bins -- then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of $k$ unknown distributions from $\mathfrak{C}.$ We analyze several natural types of distributions over $[n]$, including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems.Comment: preliminary full version of soda'13 pape

A composition theorem for parity kill number

In this work, we study the parity complexity measures ${\mathsf{C}^{\oplus}_{\min}}[f]$ and ${\mathsf{DT^{\oplus}}}[f]$. ${\mathsf{C}^{\oplus}_{\min}}[f]$ is the \emph{parity kill number} of $f$, the fewest number of parities on the input variables one has to fix in order to "kill" $f$, i.e. to make it constant. ${\mathsf{DT^{\oplus}}}[f]$ is the depth of the shortest \emph{parity decision tree} which computes $f$. These complexity measures have in recent years become increasingly important in the fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and pseudorandomness \cite{BK12, Sha11, CT13}. Our main result is a composition theorem for ${\mathsf{C}^{\oplus}_{\min}}$. The $k$-th power of $f$, denoted $f^{\circ k}$, is the function which results from composing $f$ with itself $k$ times. We prove that if $f$ is not a parity function, then ${\mathsf{C}^{\oplus}_{\min}}[f^{\circ k}] \geq \Omega({\mathsf{C}_{\min}}[f]^{k}).$ In other words, the parity kill number of $f$ is essentially supermultiplicative in the \emph{normal} kill number of $f$ (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity complexity measures of $\mathsf{Sort}^{\circ k}$ and $\mathsf{HI}^{\circ k}$. Here $\mathsf{Sort}$ is the sort function due to Ambainis \cite{Amb06}, and $\mathsf{HI}$ is Kushilevitz's hemi-icosahedron function \cite{NW95}. In doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and \cite{TWXZ13}

Fully Dynamic Min-Cut of Superconstant Size in Subpolynomial Time

We present a deterministic fully dynamic algorithm with subpolynomial worst-case time per graph update such that after processing each update of the graph, the algorithm outputs a minimum cut of the graph if the graph has a cut of size at most $c$ for some $c = (\log n)^{o(1)}$. Previously, the best update time was $\widetilde O(\sqrt{n})$ for any $c > 2$ and $c = O(\log n)$ [Thorup, Combinatorica'07].Comment: SODA 202

On the isomorphism testing of graphs

Graph Isomorphism is one of the very few classical problems in NP of unsettled complexity status. The families of highly regular structures, for example Steiner 2-designs, strongly regular graphs and primitive coherent configurations, have been perceived as difficult cases for graph isomorphism. These highly regular structures arise naturally as obstacles for both the classical group theory and combinatorial approaches for the graph isomorphism problem. In this thesis we investigate the isomorphism problem of highly regular structures. We present new results to understand the combinatorial structure of highly regular structures, and propose some new algorithms to compute the canonical forms (and thus isomorphism testing) of highly regular structures based on the structural theorems. We also give an algorithm solving the isomorphism problem of two unknown graphs in the property testing setting. Our new algorithm has sample complexity matching the information theoretical lower bound up to some multiplicative subpolynomial factor

Dynamic Kernel Sparsifiers

A geometric graph associated with a set of points $P= \{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$ and a fixed kernel function $\mathsf{K}:\mathbb{R}^d\times \mathbb{R}^d\to\mathbb{R}_{\geq 0}$ is a complete graph on $P$ such that the weight of edge $(x_i, x_j)$ is $\mathsf{K}(x_i, x_j)$. We present a fully-dynamic data structure that maintains a spectral sparsifier of a geometric graph under updates that change the locations of points in $P$ one at a time. The update time of our data structure is $n^{o(1)}$ with high probability, and the initialization time is $n^{1+o(1)}$. Under certain assumption, we can provide a fully dynamic spectral sparsifier with the robostness to adaptive adversary. We further show that, for the Laplacian matrices of these geometric graphs, it is possible to maintain random sketches for the results of matrix vector multiplication and inverse-matrix vector multiplication in $n^{o(1)}$ time, under updates that change the locations of points in $P$ or change the query vector by a sparse difference