Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth

We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our result generalizes Babai's quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur

Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits

We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following: 1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^{Omega(n)} size. 2. Any multilinear depth three circuit computing IMM_{n,d} (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^{Omega(d)} size. IMM_{n,d} can be easily computed by a poly(n,d) sized ROABP. 3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^{Omega(d)}. This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field

Towards Blackbox Identity Testing of Log-Variate Circuits

Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC\u2718) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few variables, eg. logarithmic in the size s. We give the first poly(s)-time blackbox identity test for n=O(log s) variate size-s circuits that have poly(s)-dimensional partial derivative space; eg. depth-3 diagonal circuits (or Sigma wedge Sigma^n). The former model is well-studied (Nisan,Wigderson, FOCS\u2795) but no poly(s2^n)-time identity test was known before us. We introduce the concept of cone-closed basis isolation and prove its usefulness in studying log-variate circuits. It subsumes the previous notions of rank-concentration studied extensively in the context of ROABP models

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

The orbit of an n-variate polynomial f(?) over a field ? is the set {f(A?+?) : A ? GL(n,?) and ? ? ??}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: 1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials. 2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ? ?}, which is complete for arithmetic formulas. 3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials. 4) Polynomials computable by occur-once formulas

The Power of Natural Properties as Oracles

We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that ZPEXP^{MCSP} !subseteq P/poly, which should be contrasted with the previously known circuit lower bound ZPEXP^{NP} !subseteq P/poly. We also show that, assuming the existence of Indistinguishability Obfuscators (IO), SAT and MCSP are equivalent in the sense that one has a ZPP algorithm if and only the other one does. We interpret our results as providing some evidence that MCSP may be NP-hard under randomized polynomial-time reductions

Isomorphism Testing for Graphs Excluding Small Minors

We prove that there is a graph isomorphism test running in time $n^{\operatorname{polylog}(h)}$ on $n$-vertex graphs excluding some $h$-vertex graph as a minor. Previously known bounds were $n^{\operatorname{poly}(h)}$ (Ponomarenko, 1988) and $n^{\operatorname{polylog}(n)}$ (Babai, STOC 2016). For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments