Efficient Black-Box Identity Testing for Free Group Algebras

Hrubes and Wigderson [Pavel Hrubes and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n

On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials

We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the rectangular symbolic matrix in both commutative and noncommutative settings. The main results are: - We show an explicit O^*(binom{n}{downarrow k/2})-size ABP construction for noncommutative permanent polynomial of k x n symbolic matrix. We obtain this via an explicit ABP construction of size O^*(binom{n}{downarrow k/2}) for S_{n,k}^*, noncommutative symmetrized version of the elementary symmetric polynomial S_{n,k}. - We obtain an explicit O^*(2^k)-size ABP construction for the commutative rectangular determinant polynomial of the k x n symbolic matrix. - In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard

Fast Exact Algorithms Using Hadamard Product of Polynomials

Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f in F[X], where X={x_1,x_2,...,x_n} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [Ioannis Koutis, 2008; Ryan Williams, 2009; Ioannis Koutis and Ryan Williams, 2016]. - (k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f. - k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f. Our algorithms are based on the fact that the Hadamard product f o S_{n,k}, is the degree-k multilinear part of f, where S_{n,k} is the k^{th} elementary symmetric polynomial. - For (k,n)-MLC problem, we give a deterministic algorithm of run time O^*(n^(k/2+c log k)) (where c is a constant), answering an open question of Koutis and Williams [Ioannis Koutis and Ryan Williams, 2016]. As corollaries, we show O^*(binom{n}{downarrow k/2})-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching. - For k-MMD problem, we give a randomized algorithm of run time 4.32^k * poly(n,k). Our algorithm uses only poly(n,k) space. This matches the run time of a recent algorithm [Cornelius Brand et al., 2018] for k-MMD which requires exponential (in k) space. Other results include fast deterministic algorithms for (k,n)-MLC and k-MMD problems for depth three circuits

Equivalence Testing of Weighted Automata over Partially Commutative Monoids

Motivated by equivalence testing of k-tape automata, we study the equivalence testing of weighted automata in the more general setting, over partially commutative monoids (in short, pc monoids), and show efficient algorithms in some special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the edge clique cover number of the non-commutation graph of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm for equivalence testing. As a corollary, we obtain the first deterministic quasi-polynomial time algorithms for equivalence testing of k-tape weighted automata and for equivalence testing of deterministic k-tape automata for constant k. Prior to this, the best complexity upper bound for these k-tape automata problems were randomized polynomial-time, shown by Worrell [James Worrell, 2013]. Finding a polynomial-time deterministic algorithm for equivalence testing of deterministic k-tape automata for constant k has been open for several years [Emily P. Friedman and Sheila A. Greibach, 1982] and our results make progress. We also consider pc monoids for which the non-commutation graphs have an edge cover consisting of at most k cliques and star graphs for any constant k. We obtain a randomized polynomial-time algorithm for equivalence testing of weighted automata over such monoids. Our results are obtained by designing efficient zero-testing algorithms for weighted automata over such pc monoids

Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Let F[X] be the polynomial ring over the variables X={x_1,x_2, ..., x_n}. An ideal I= generated by univariate polynomials {p_i(x_i)}_{i=1}^n is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results. - Let f(X) in F[l_1, ..., l_r] be a (low rank) polynomial given by an arithmetic circuit where l_i : 1 be a univariate ideal. Given alpha in F^n, the (unique) remainder f(X) mod I can be evaluated at alpha in deterministic time d^{O(r)} * poly(n), where d=max {deg(f),deg(p_1)...,deg(p_n)}. This yields a randomized n^{O(r)} algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n^{O(r)} algorithm for evaluating the permanent of a n x n matrix of rank r, over any field F. Over Q, an algorithm of similar run time for low rank permanent is due to Barvinok [Barvinok, 1996] via a different technique. - Let f(X)in F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I=. We show that in the special case when I=, we obtain a randomized O^*(4.08^k) algorithm that uses poly(n,k) space. - Given f(X)in F[X] by an arithmetic circuit and I=, membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I=

Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings

In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring F{X}. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson, and Yehudayoff, and they showed lower bounds and proved completeness results. We consider Polynomial Identity Testing and Polynomial Factorization in F{X} and show the following results. 1. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give a deterministic polynomial algorithm to decide if f is identically zero. Our result is obtained by a suitable adaptation of the PIT algorithm of Raz and Shpilka for noncommutative ABPs. 2. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give an efficient deterministic algorithm to compute circuits for the irreducible factors of f in polynomial time when F is the field of rationals. Over finite fields of characteristic p, our algorithm runs in time polynomial in input size and p