arXiv subjects

V. Arvind

Publications and source records attributed to V. Arvind.

At least 19 recordsLinked to original sources

arXiv2025

Derandomizing Isolation In Catalytic Logspace

A language is said to be in catalytic logspace if we can test membership using a deterministic logspace machine that has an additional read/write tape filled with arbitrary data whose contents have to be restored to their original value at the end of the computation. The model of catalytic computation was introduced by Buhrman et al [STOC2014]. As our first result, we obtain a catalytic logspace algorithm for computing a minimum weight witness to a search problem, with small weights, provided the algorithm is given oracle access for the corresponding weighted decision problem. In particular, our reduction yields CL algorithms for the search versions of the following three problems: planar perfect matching, planar exact perfect matching and weighted arborescences in weighted digraphs. Our second set of results concern the significantly larger class CL^{NP}_{2-round}. We show that CL^{NP}_{2-round} contains SearchSAT and the complexity classes BPP, MA and ZPP^{NP[1]}. While SearchSAT is shown to be in CL^{NP}_{2-round} using the isolation lemma, the other three containments, while based on the compress-or-random technique, use the Nisan-Wigderson [JCSS 1994] based pseudo-random generator. These containments show that CL^{NP}_{2-round} resembles ZPP^NP more than P^{NP}, providing some weak evidence that CL is more like ZPP than P. For our third set of results we turn to isolation well inside catalytic classes. We consider the unambiguous catalytic class CTISP[poly(n),logn,log^2n]^UL and show that it contains reachability and therefore NL. This is a catalytic version of the result of van Melkebeek & Prakriya [SIAM J. Comput. 2019]. Building on their result, we also show a tradeoff between the workspace of the oracle and the catalytic space of the base machine. Finally, we extend these catalytic upper bounds to LogCFL.

cs.CC↗

arXiv2025

Fast exact algorithms via the Matrix Tree Theorem

Fast exact algorithms are known for Hamiltonian paths in undirected and directed bipartite graphs through elegant though involved algorithms that are quite different from each other. We devise algorithms that are simple and similar to each other while having the same upper bounds. The common features of these algorithms is the use of the Matrix-Tree theorem and sieving using roots of unity. Next, we use the framework to provide alternative algorithms to count perfect matchings in bipartite graphs on $n$ vertices, i.e., computing the $\{0,1\}$-permanent of a square $n/2 \times n/2$ matrix which runs in a time similar to Ryser. We demonstrate the flexibility of our method by counting the number of ways to vertex partition the graph into $k$-stars (a $k$-star consist of a tree with a root having $k-1$ children that are all leaves). Interestingly, our running time improves to $O^*((1+\epsilon_k)^n)$ with $\epsilon_k \rightarrow 0$ as $k \rightarrow \infty$. As an aside, making use of Bj\"orklund's algorithm for exact counting perfect matchings in general graphs, we show that the count of maximum matchings can be computed in time $O^*(2^\nu)$ where $\nu$ is the size of a maximum matching. The crucial ingredient here is the famous Gallai-Edmonds decomposition theorem. All our algorithms run in polynomial space.

cs.DS↗

arXiv2024

Revisiting Tree Canonization using polynomials

Graph Isomorphism (GI) is a fundamental algorithmic problem. Amongst graph classes for which the computational complexity of GI has been resolved, trees are arguably the most fundamental. Tree Isomorphism is complete for deterministic logspace, a tiny subclass of polynomial time, by Lindell's result. Over three decades ago, he devised a deterministic logspace algorithm that computes a string which is a canon for the input tree -- two trees are isomorphic precisely when their canons are identical. Inspired by Miller-Reif's reduction of Tree Isomorphism to Polynomial Identity Testing, we present a new logspace algorithm for tree canonization fundamentally different from Lindell's algorithm. Our algorithm computes a univariate polynomial as canon for an input tree, based on the classical Eisenstein's criterion for the irreducibility of univariate polynomials. This can be implemented in logspace by invoking the well known Buss et al. algorithm for arithmetic formula evaluation. However, we have included in the appendix a simpler self-contained proof showing that arithmetic formula evaluation is in logspace. This algorithm is conceptually very simple, avoiding the delicate case analysis and complex recursion that constitute the core of Lindell's algorithm. We illustrate the adaptability of our algorithm by extending it to a couple of other classes of graphs.

cs.CC↗

arXiv2024

On the Expressibility of the Reconstructional Color Refinement

One of the most basic facts related to the famous Ulam reconstruction conjecture is that the connectedness of a graph can be determined by the deck of its vertex-deleted subgraphs, which are considered up to isomorphism. We strengthen this result by proving that connectedness can still be determined when the subgraphs in the deck are given up to equivalence under the color refinement isomorphism test. Consequently, this implies that connectedness is recognizable by Reconstruction Graph Neural Networks, a recently introduced GNN architecture inspired by the reconstruction conjecture (Cotta, Morris, Ribeiro 2021).

cs.CC↗

arXiv2024

Trading Determinism for Noncommutativity in Edmonds' Problem

Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x\in X_i$ commute with the variables $x'\in X_j$. Given as input a square matrix $T$ whose entries are linear forms over $\mathbb{Q}\langle{X}\rangle$, we consider the problem of checking if $T$ is invertible or not over the universal skew field of fractions of the partially commutative polynomial ring $\mathbb{Q}\langle{X}\rangle$ [Klep-Vinnikov-Volcic (2020)]. In this paper, we design a deterministic polynomial-time algorithm for this problem for constant $k$. The special case $k=1$ is the noncommutative Edmonds' problem (NSINGULAR) which has a deterministic polynomial-time algorithm by recent results [Garg-Gurvits-Oliveira-Wigderson (2016), Ivanyos-Qiao-Subrahmanyam (2018), Hamada-Hirai (2021)]. En-route, we obtain the first deterministic polynomial-time algorithm for the equivalence testing problem of $k$-tape \emph{weighted} automata (for constant $k$) resolving a long-standing open problem [Harju and Karhum"{a}ki(1991), Worrell (2013)]. Algebraically, the equivalence problem reduces to testing whether a partially commutative rational series over the partitioned set $X$ is zero or not [Worrell (2013)]. Decidability of this problem was established by Harju and Karhum\"{a}ki (1991). Prior to this work, a \emph{randomized} polynomial-time algorithm for this problem was given by Worrell (2013) and, subsequently, a deterministic quasipolynomial-time algorithm was also developed [Arvind et al. (2021)].

cs.CC↗

arXiv2023

On a Hierarchy of Spectral Invariants for Graphs

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by F\"urer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to F\"urer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

cs.CC↗

arXiv2023

Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time

Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017). Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, Mukhopadhyay (2022)]. In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

cs.CC↗

arXiv2023

The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem

Let $G$ be a finite group given as input by its multiplication table. For a subset $S$ of $G$ and an element $g\in G$ the Cayley Group Membership Problem (denoted CGM) is to check if $g$ belongs to the subgroup generated by $S$. While this problem is easily seen to be in polynomial time, pinpointing its parallel complexity has been of research interest over the years. In this paper we further explore the parallel complexity of the abelian CGM problem, with focus on the dynamic setting: the generating set $S$ changes with insertions and deletions and the goal is to maintain a data structure that supports efficient membership queries to the subgroup $\angle{S}$. We obtain the following results: 1. We first consider the more general problem of Monoid Membership. When $G$ is a commutative monoid we give a deterministic dynamic algorithm constant time parallel algorithm for membership testing that supports $O(1)$ insertions and deletions in each step. 2. Building on the previous result we show that there is a dynamic randomized constant-time parallel algorithm for abelian CGM that supports polylogarithmically many insertions/deletions to $S$ in each step. 3. If the number of insertions/deletions is at most $O(\log n/\log\log n)$ then we obtain a deterministic dynamic constant-time parallel algorithm for the problem. 4. We obtain analogous results for the dynamic abelian Group Isomorphism.

cs.CC↗

arXiv2023

On Lifting Lower Bounds for Noncommutative Circuits using Automata

We revisit the main result of Carmosino et al \cite{CILM18} which shows that an $\Omega(n^{\omega/2+\epsilon})$ size noncommutative arithmetic circuit size lower bound (where $\omega$ is the matrix multiplication exponent) for a constant-degree $n$-variate polynomial family $(g_n)_n$, where each $g_n$ is a noncommutative polynomial, can be ``lifted'' to an exponential size circuit size lower bound for another polynomial family $(f_n)$ obtained from $(g_n)$ by a lifting process. In this paper, we present a simpler and more conceptual automata-theoretic proof of their result.

cs.CC↗

arXiv2023

Aspects of the commuting graph

The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given graph is the commuting graph of a finite group; we give a quasipolynomial algorithm, and a polynomial algorithm for the case when the group is an extra\-special p-group for p an odd prime; we give new results on the question of whether the commuting graph of a given group is a cograph or a chordal graph, two classes of graphs defined by forbidden subgraphs. The problems are not unrelated, since there are a number of cases where hard computational problems on graphs are easier when restricted to special classes of graphs; we conjecture that the recognition problem is polynomial for cographs and chordal graphs.

math.GR↗

arXiv2023

Multivariate to Bivariate Reduction for Noncommutative Polynomial Factorization

Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following: (1) In the white-box setting, given an n-variate noncommutative polynomial f in F over a field F (either a finite field or the rationals) as an arithmetic circuit (or algebraic branching program), computing a complete factorization of f is deterministic polynomial-time reducible to white-box factorization of a noncommutative bivariate polynomial g in F ; the reduction transforms f into a circuit for g (resp. ABP for g), and given a complete factorization of g the reduction recovers a complete factorization of f in polynomial time. We also obtain a similar deterministic polynomial-time reduction in the black-box setting. (2) Additionally, we show over the field of rationals that bivariate linear matrix factorization of 4 x 4 matrices is at least as hard as factoring square-free integers. This indicates that reducing noncommutative polynomial factorization to linear matrix factorization (as done in our recent work [AJ22]) is unlikely to succeed over the field of rationals even in the bivariate case. In contrast, multivariate linear matrix factorization for 3 x 3 matrices over rationals is in polynomial time.

cs.CC↗

arXiv2022

On Identity Testing and Noncommutative Rank Computation over the Free Skew Field

The identity testing of rational formulas (RIT) in the free skew field efficiently reduces to computing the rank of a matrix whose entries are linear polynomials in noncommuting variables\cite{HW15}. This rank computation problem has deterministic polynomial-time white-box algorithms \cite{GGOW16, IQS18} and a randomized polynomial-time algorithm in the black-box setting \cite{DM17}. In this paper, we propose a new approach for efficient derandomization of \emph{black-box} RIT. Additionally, we obtain results for matrix rank computation over the free skew field, and construct efficient linear pencil representations for a new class of rational expressions. More precisely, we show the following results: 1. Under the hardness assumption that the ABP (algebraic branching program) complexity of every polynomial identity for the $k\times k$ matrix algebra is $2^{\Omega(k)}$ \cite{BW05}, we obtain a subexponential-time black-box algorithm for RIT in almost general setting. This can be seen as the first "hardness implies derandomization" type theorem for rational formulas. 2. We show that the noncommutative rank of any matrix over the free skew field whose entries have small linear pencil representations can be computed in deterministic polynomial time. Prior to this, an efficient rank computation was only known for matrices with noncommutative formulas as entries\cite{GGOW20}. As special cases of our algorithm, we obtain the first deterministic polynomial-time algorithms for rank computation of matrices whose entries are noncommutative ABPs or rational formulas. 3. Motivated by the definition given by Bergman\cite{Ber76}, we define a new class that contains noncommutative ABPs and rational formulas. We obtain a polynomial-size linear pencil representation for this class. As a by-product, we obtain a white-box deterministic polynomial-time identity testing algorithm for the class.

cs.CC↗

arXiv2022

Recognizing the Commuting Graph of a Finite Group

In this paper we study the realizability question for commuting graphs of finite groups: Given an undirected graph $X$ is it the commuting graph of a group $G$? And if so, to determine such a group. We seek efficient algorithms for this problem. We make some general observations on this problem, and obtain a polynomial-time algorithm for the case of extraspecial groups.

math.GR↗

arXiv2022

Matrix Polynomial Factorization via Higman Linearization

In continuation to our recent work on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results. (1) Given as input a full rank $d\times d$ matrix $M$ whose entries $M_{ij}$ are polynomials in the free noncommutative ring $\mathbb{F}_q\langle x_1,x_2,\ldots,x_n \rangle$, where each $M_{ij}$ is given by a noncommutative arithmetic formula of size at most $s$, we give a randomized algorithm that runs in time polynomial in $d,s, n$ and $\log_2q$ that computes a factorization of $M$ as a matrix product $M=M_1M_2\cdots M_r$, where each $d\times d$ matrix factor $M_i$ is irreducible (in a well-defined sense) and the entries of each $M_i$ are polynomials in $\mathbb{F}_q \langle x_1,x_2,\ldots,x_n \rangle$ that are output as algebraic branching programs. We also obtain a deterministic algorithm for the problem that runs in $poly(d,n,s,q)$. (2)A special case is the efficient factorization of matrices whose entries are univariate polynomials in $\mathbb{F}[x]$. When $\mathbb{F}$ is a finite field the above result applies. When $\mathbb{F}$ is the field of rationals we obtain a deterministic polynomial-time algorithm for the problem.

cs.CC↗

arXiv2022

On Efficient Noncommutative Polynomial Factorization via Higman Linearization

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F of polynomials in noncommuting variables x1,x2,...,xn over the field F. We obtain the following result Given a noncommutative algebraic branching program of size s computing a noncommutative polynomial f in F as input, where F=Fq is a finite field, we give a randomized algorithm that runs in time polynomial in s, n and log q that computes a factorization of the polynomial f as a product f=f1f2...fr, where each fi is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming the given algebraic branching program computing f into a linear matrix L using Higman's linearization of polynomials. We then factorize the linear matrix L and recover the factorization of the polynomial f. We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.

cs.CC↗

arXiv2022

Black-box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial-time

Hrube\v{s} and Wigderson (2015) initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, deterministic polynomial-time algorithms are known for this problem following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and Ivanyos, Qiao, and Subrahmanyam (2018). A central open problem in this area is to design efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this problem for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including key concepts from matrix coefficient realization theory (Vol\v{c}i\v{c}, 2018) and properties of cyclic division algebra (Lam, 2001). En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra of small index.

cs.CC↗

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 the Weisfeiler-Leman Dimension of Fractional Packing

The $k$-dimensional Weisfeiler-Leman procedure ($k$-WL), which colors $k$-tuples of vertices in rounds based on the neighborhood structure in the graph, has proven to be immensely fruitful in the algorithmic study of Graph Isomorphism. More generally, it is of fundamental importance in understanding and exploiting symmetries in graphs in various settings. Two graphs are $k$-WL-equivalent if the $k$-dimensional Weisfeiler-Leman procedure produces the same final coloring on both graphs. 1-WL-equivalence is known as fractional isomorphism of graphs, and the $k$-WL-equivalence relation becomes finer as $k$ increases. We investigate to what extent standard graph parameters are preserved by $k$-WL-equivalence, focusing on fractional graph packing numbers. The integral packing numbers are typically NP-hard to compute, and we discuss applicability of $k$-WL-invariance for estimating the integrality gap of the LP relaxation provided by their fractional counterparts.

cs.CC↗