arXiv subjects

V. Arvind

Publications and source records attributed to V. Arvind.

At least 37 records · Page 2Linked to original sources

arXiv2020

Equivalence Testing of Weighted Automata over Partially Commutative Monoids

We study \emph{multiplicity equivalence} testing of automata over partially commutative monoids (pc monoids) and show efficient algorithms in special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the clique cover number of the non-commutation graph (the minimum number of cliques covering the graph) of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm. As a consequence, we also obtain the first deterministic quasi-polynomial time algorithms for multiplicity equivalence testing of $k$-tape automata and for equivalence testing of deterministic $k$-tape automata for constant $k$. Prior to this, a randomized polynomial-time algorithm for the above problems was shown by Worrell [ICALP 2013]. We also consider pc monoids for which the non-commutation graphs have cover consisting of at most $k$ cliques and star graphs for any constant $k$. We obtain randomized polynomial-time algorithm for multiplicity equivalence testing of automata over such monoids.

cs.FL↗

arXiv2019

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 \emph{rectangular} symbolic matrix in both commutative and noncommutative settings. The main results are: 1. We show an explicit $O^{*}({n\choose {\downarrow k/2}})$-size ABP construction for noncommutative permanent polynomial of $k\times n$ symbolic matrix. We obtain this via an explicit ABP construction of size $O^{*}({n\choose {\downarrow k/2}})$ for $S_{n,k}^*$, noncommutative symmetrized version of the elementary symmetric polynomial $S_{n,k}$. 2. We obtain an explicit $O^{*}(2^k)$-size ABP construction for the commutative rectangular determinant polynomial of the $k\times n$ symbolic matrix. 3. In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is $W[1]$-hard.

cs.CC↗

arXiv2019

Efficient Black-Box Identity Testing over Free Group Algebra

Hrubeš and Wigderson [HW14] 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. It is now known that the problem can be solved in deterministic polynomial time in the white-box model for noncommutative formulas with inverses, and in randomized polynomial time in the black-box model [GGOW16, IQS18, DM18], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions remains open in general (when the formula size is not polynomially bounded). We solve the problem for a natural special case. We consider polynomial expressions in the free group algebra $\mathbb{F}\langle X, X^{-1}\rangle$ where $X=\{x_1, x_2, \ldots, x_n\}$, a subclass of rational expressions of inversion height one. Our main results are the following. 1. Given a degree $d$ expression $f$ in $\mathbb{F}\langle X, X^{-1}\rangle$ as a black-box, we obtain a randomized $\text{poly}(n,d)$ algorithm to check whether $f$ is an identically zero expression or not. We obtain this by generalizing the Amitsur-Levitzki theorem [AL50] to $\mathbb{F}\langle X, X^{-1}\rangle$. 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 $\mathbb{F}\langle X, X^{-1}\rangle$ of degree at most $D$, and sparsity $s$, as black-box, we can check whether $f$ is identically zero or not in randomized $\text{poly}(n,\log s, \log D)$ time.

cs.CC↗

arXiv2019

On Weisfeiler-Leman Invariance: Subgraph Counts and Related Graph Properties

The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a fruitful approach to the Graph Isomorphism problem. 2-WL corresponds to the original algorithm suggested by Weisfeiler and Leman over 50 years ago. 1-WL is the classical color refinement routine. Indistinguishability by $k$-WL is an equivalence relation on graphs that is of fundamental importance for isomorphism testing, descriptive complexity theory, and graph similarity testing which is also of some relevance in artificial intelligence. Focusing on dimensions $k=1,2$, we investigate subgraph patterns whose counts are $k$-WL invariant, and whose occurrence is $k$-WL invariant. We achieve a complete description of all such patterns for dimension $k=1$ and considerably extend the previous results known for $k=2$.

cs.DM↗

arXiv2018

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

Let $\mathbb{F}[X]$ be the polynomial ring over the variables $X=\{x_1,x_2, \ldots, x_n\}$. An ideal $I=\langle p_1(x_1), \ldots, p_n(x_n)\rangle$ generated by univariate polynomials $\{p_i(x_i)\}_{i=1}^n$ is a \emph{univariate ideal}. We study the ideal membership problem for the univariate ideals and show the following results. \item Let $f(X)\in\mathbb{F}[\ell_1, \ldots, \ell_r]$ be a (low rank) polynomial given by an arithmetic circuit where $\ell_i : 1\leq i\leq r$ are linear forms, and $I=\langle p_1(x_1), \ldots, p_n(x_n)\rangle$ be a univariate ideal. Given $\vecα\in {\mathbb{F}}^n$, the (unique) remainder $f(X) \pmod I$ can be evaluated at $\vecα$ in deterministic time $d^{O(r)}\cdot poly(n)$, where $d=\max\{°(f),°(p_1)\ldots,°(p_n)\}$. This yields an $n^{O(r)}$ 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 $n\times n$ matrix of rank $r$, over any field $\mathbb{F}$. Over $\mathbb{Q}$, an algorithm of similar run time for low rank permanent is due to Barvinok[Bar96] via a different technique. \item Let $f(X)\in\mathbb{F}[X]$ be given by an arithmetic circuit of degree $k$ ($k$ treated as fixed parameter) and $I=\langle p_1(x_1), \ldots, p_n(x_n)\rangle$. We show in the special case when $I=\langle x_1^{e_1}, \ldots, x_n^{e_n}\rangle$, we obtain a randomized $O^*(4.08^k)$ algorithm that uses $poly(n,k)$ space. \item Given $f(X)\in\mathbb{F}[X]$ by an arithmetic circuit and $I=\langle p_1(x_1), \ldots, p_k(x_k) \rangle$, membership testing is $W[1]$-hard, parameterized by $k$. The problem is $MINI[1]$-hard in the special case when $I=\langle x_1^{e_1}, \ldots, x_k^{e_k}\rangle$.

cs.DS↗

arXiv2018

A Note on Polynomial Identity Testing for Depth-3 Circuits

Let $C$ be a depth-3 arithmetic circuit of size at most $s$, computing a polynomial $ f \in \mathbb{F}[x_1,\ldots, x_n] $ (where $\mathbb{F}$ = $\mathbb{Q}$ or $\mathbb{C}$) and the fan-in of the product gates of $C$ is bounded by $d$. We give a deterministic polynomial identity testing algorithm to check whether $f\equiv 0$ or not in time $ 2^d \text{ poly}(n,s) $.

cs.CC↗

arXiv2017

Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs $X_1=(V,E_1)$ and $X_2=(V,E_2)$ and a subset $R\subseteq V\times V$ of forbidden pairs whether there is an isomorphism $π$ from $X_1$ to $X_2$ such that $π(i)\neq j$ for all $(i,j)\in R$. We prove that this problem and several of its generalizations are in fact in FPT: - The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of $R$ as parameters. The problem remains in FPT if a CNF of such constraints is allowed. It follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017]. - When the weight and complexity are unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle. - Checking if there is an isomorphism $π$ between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that $π$ can be expressed as a product of t transpositions. - We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT.

cs.CC↗

arXiv2017

Efficient Identity Testing and Polynomial Factorization over Non-associative Free Rings

In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring $\mathbb{F}\{x_1,x_2,\ldots,x_n\}$. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson, and Yehudayoff [HWY10], and they showed lower bounds and proved completeness results. We consider Polynomial Identity Testing (PIT) and polynomial factorization over $\mathbb{F}\{x_1,x_2,\ldots,x_n\}$ and show the following results. (1) Given an arithmetic circuit $C$ of size $s$ computing a polynomial $f\in \mathbb{F} \{x_1,x_2,\ldots,x_n\}$ of degree $d$, we give a deterministic $poly(n,s,d)$ algorithm to decide if $f$ is identically zero polynomial or not. Our result is obtained by a suitable adaptation of the PIT algorithm of Raz-Shpilka [RS05] for noncommutative ABPs. (2) Given an arithmetic circuit $C$ of size $s$ computing a polynomial $f\in \mathbb{F} \{x_1,x_2,\ldots,x_n\}$ of degree $d$, we give an efficient deterministic algorithm to compute circuits for the irreducible factors of $f$ in time $poly(n,s,d)$ when $\mathbb{F}=\mathbb{Q}$. Over finite fields of characteristic $p$, our algorithm runs in time $poly(n,s,d,p)$.

cs.CC↗

arXiv2016

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the complexity of the following problems: Given a colored graph X=(V,E,c), compute a minimum cardinality set S of vertices such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G on [n] given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT algorithms. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c) compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time (even in logspace), while starting from color class size 4 they become W[P]-hard.

cs.CC↗

arXiv2016

Randomized Polynomial Time Identity Testing for Noncommutative Circuits

In this paper we show that the black-box polynomial identity testing for noncommutative polynomials $f\in\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$ of degree $D$ and sparsity $t$, can be done in randomized $\poly(n,\log t,\log D)$ time. As a consequence, if the black-box contains a circuit $C$ of size $s$ computing $f\in\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$ which has at most $t$ non-zero monomials, then the identity testing can be done by a randomized algorithm with running time polynomial in $s$ and $n$ and $\log t$. This makes significant progress on a question that has been open for over ten years. The earlier result by Bogdanov and Wee [BW05], using the classical Amitsur-Levitski theorem, gives a randomized polynomial-time algorithm only for circuits of polynomially bounded syntactic degree. In our result, we place no restriction on the degree of the circuit. Our algorithm is based on automata-theoretic ideas introduced in [AMS08,AM08]. In those papers, the main idea was to construct deterministic finite automata that isolate a single monomial from the set of nonzero monomials of a polynomial $f$ in $\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$. In the present paper, since we need to deal with exponential degree monomials, we carry out a different kind of monomial isolation using nondeterministic automata.

cs.CC↗

arXiv2015

Some Lower Bound Results for Set-Multilinear Arithmetic Computations

In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to show lower bound results. Some of our results extend existing lower bounds, while others are new and raise open questions. More specifically, our main results are the following: (1) We observe that set-multilinear arithmetic circuits can be transformed into shallow set-multilinear circuits efficiently, similar to depth reduction results of [VSBR83,RY08] for more general commutative circuits. As a consequence, we note that polynomial size set-multilinear circuits have quasi-polynomial size set-multilinear branching programs. We show that \emph{narrow} set-multilinear ABPs (with a restricted number of set types) computing the Permanent polynomial $\mathrm{PER}_n$ require $2^{n^{Ω(1)}}$ size. A similar result for general set-multilinear ABPs appears difficult as it would imply that the Permanent requires superpolynomial size set-multilinear circuits. It would also imply that the noncommutative Permanent requires superpolynomial size noncommutative arithmetic circuits. (2) Indeed, we also show that set-multilinear branching programs are exponentially more powerful than \emph{interval} multilinear circuits (where the index sets for each gate is restricted to be an interval w.r.t.\ some ordering), assuming the sum-of-squares conjecture. This further underlines the power of set-multilinear branching programs. (3) Finally, we consider set-multilinear circuits with restrictions on the number of proof trees of monomials computed by it, and prove exponential lower bounds results. This raises some new lower bound questions.

cs.CC↗

arXiv2015

Noncommutative Valiant's Classes: Structure and Complete Problems

In this paper we explore the noncommutative analogues, $\mathrm{VP}_{nc}$ and $\mathrm{VNP}_{nc}$, of Valiant's algebraic complexity classes and show some striking connections to classical formal language theory. Our main results are the following: (1) We show that Dyck polynomials (defined from the Dyck languages of formal language theory) are complete for the class $\mathrm{VP}_{nc}$ under $\le_{abp}$ reductions. Likewise, it turns out that $\mathrm{PAL}$ (Palindrome polynomials defined from palindromes) are complete for the class $\mathrm{VSKEW}_{nc}$ (defined by polynomial-size skew circuits) under $\le_{abp}$ reductions. The proof of these results is by suitably adapting the classical Chomsky-Schützenberger theorem showing that Dyck languages are the hardest CFLs. (2) Next, we consider the class $\mathrm{VNP}_{nc}$. It is known~\cite{HWY10a} that, assuming the sum-of-squares conjecture, the noncommutative polynomial $\sum_{w\in\{x_0,x_1\}^n}ww$ requires exponential size circuits. We unconditionally show that $\sum_{w\in\{x_0,x_1\}^n}ww$ is not $\mathrm{VNP}_{nc}$-complete under the projection reducibility. As a consequence, assuming the sum-of-squares conjecture, we exhibit a strictly infinite hierarchy of p-families under projections inside $\mathrm{VNP}_{nc}$ (analogous to Ladner's theorem~\cite{Ladner75}). In the final section we discuss some new $\mathrm{VNP}_{nc}$-complete problems under $\le_{abp}$-reductions. (3) Inside $\mathrm{VP}_{nc}$ too we show there is a strict hierarchy of p-families (based on the nesting depth of Dyck polynomials) under the $\le_{abp}$ reducibility.

cs.CC↗

arXiv2015

Graph Isomorphism, Color Refinement, and Compactness

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if it succeeds in distinguishing G from any non-isomorphic graph H. Tinhofer (1991) explored a linear programming approach to Graph Isomorphism and defined compact graphs: A graph is compact if its fractional automorphisms polytope is integral. Tinhofer noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing and recognizing compact graphs in polynomial time remains an open question. Our results are summarized below: - We show that amenable graphs are recognizable in time O((n + m)logn), where n and m denote the number of vertices and the number of edges in the input graph. - We show that all amenable graphs are compact. - We study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. The corresponding classes of graphs form a hierarchy and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.

cs.CC↗

arXiv2015

On the Complexity of Noncommutative Polynomial Factorization

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring $\mathbb{F}\langle x_1,x_2,\dots,x_n\rangle$ of polynomials over the field $\mathbb{F}$ and noncommuting variables $x_1,x_2,\ldots,x_n$. Our main results are the following. Although $\mathbb{F}\langle x_1,x_2,\dots,x_n \rangle$ is not a unique factorization ring, we note that variable-disjoint factorization in $\mathbb{F}\langle x_1,x_2,\dots,x_n \rangle$ has the uniqueness property. Furthermore, we prove that computing the variable-disjoint factorization is polynomial-time equivalent to Polynomial Identity Testing (both when the input polynomial is given by an arithmetic circuit or an algebraic branching program). We also show that variable-disjoint factorization in the black-box setting can be efficiently computed (where the factors computed will be also given by black-boxes, analogous to the work [KT91] in the commutative setting). As a consequence of the previous result we show that homogeneous noncommutative polynomials and multilinear noncommutative polynomials have unique factorizations in the usual sense, which can be efficiently computed. Finally, we discuss a polynomial decomposition problem in $\mathbb{F}\langle x_1,x_2,\dots,x_n\rangle$ which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it.

cs.CC↗

arXiv2012

Near-Optimal Expanding Generating Sets for Solvable Permutation Groups

Let $G = $ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n^2)$ for $G$. More precisely, the algorithm computes a subset $T\subset G$ of size $\tilde{O}(n^2)(1/λ)^{O(1)}$ such that the undirected Cayley graph $Cay(G,T)$ is a $λ$-spectral expander (the $\tilde{O}$ notation suppresses $\log ^{O(1)}n$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n\poly(\log d))(\frac{1}{\varepsilon})^{O(1)}$ for the groups $\Z_d^n$. The earlier known size bound was $O((d+n/\varepsilon^2))^{11/2}$ given by \cite{AMN98}.

cs.CC↗

arXiv2009

On the hardness of the noncommutative determinant

In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below: 1. We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent. Consequently, the commutative permanent polynomial has small commutative arithmetic circuits. 2. For any field F we show that computing the n X n permanent over F is polynomial-time reducible to computing the 2n X 2n (noncommutative) determinant whose entries are O(n^2) X O(n^2) matrices over the field F. 3. We also derive as a consequence that computing the n X n permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of n^{O(1)} dimension. Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials.

cs.CC↗

arXiv2009

On Lower Bounds for Constant Width Arithmetic Circuits

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of width k. It follows, from the definition of the polynomial, that the constant-width and the constant-depth hierarchies of monotone arithmetic circuits are infinite, both in the commutative and the noncommutative settings. 2. We prove hardness-randomness tradeoffs for identity testing constant-width commutative circuits analogous to [KI03,DSY08].

cs.CC↗

arXiv2009

Arithmetic Circuits and the Hadamard Product of Polynomials

Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. 1. We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class $\ceql$, and over fields of characteristic $p$ the problem is in $\ModpL/\Poly$. 2.We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as the Hadamard product of two monotone multilinear polynomials. In contrast the Permanent can be expressed as the Hadamard product of two monotone multilinear formulas of quadratic size.

cs.CC↗