In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring 𝔽 of polynomials in noncommuting variables x_1,x_2,…,x_n over the field 𝔽 . We obtain the following result: The algorithm works by first transforming f into a linear matrix L using Higman linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f . We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing a nontrivial common invariant subspace of a collection of matrices over finite fields.

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. 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.

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).

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, and 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.

We study the noncommutative rank problem, ncRANK, of computing the rank of matrices with linear entries in n noncommuting variables and the problem of noncommutative Rational Identity Testing, RIT, which is to decide if a given rational formula in n noncommuting variables is zero on its domain of definition. Motivated by the question whether these problems have deterministic NC algorithms, we revisit their interrelationship from a parallel complexity point of view. We show the following results: 1. Based on Cohn's embedding theorem we show deterministic NC reductions from multivariate ncRANK to bivariate ncRANK and from multivariate RIT to bivariate RIT. 2. We obtain a deterministic NC-Turing reduction from bivariate to bivariate ncRANK, thereby proving that a deterministic NC algorithm for bivariate ncRANK would imply that both multivariate RIT and multivariate ncRANK are in deterministic NC.

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 Karhumäki(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 Karhumäki (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)].

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.

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.

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.

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.

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.

In this paper we study the algorithmic complexity of the following problems: (1) Given a vertex-colored graph X = (V,E,c) , compute a minimum cardinality set of vertices S⊆ V 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 ≤ Sym( n ) given by generators, i.e., a minimum cardinality subset S ⊆ [ 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. (2) A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure.

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over $\F_2$ each of which is a product of affine forms. We focus on the case of k-CNF formulas (the k-SUB-SAT problem). Clearly, it is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show NP-hardness for k=2 and W[1]-hardness parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-2-SUB-SAT is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances. On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with runtime 1.5^r for 2-SUB-SAT, where $r$ is the subspace dimension and an O^*(1.4312)^n time algorithm where $n$ is the number of variables. For k more than 2, while known algorithms for solving a system of degree $k$ polynomial equations already imply a solution with runtime 2^{r(1-1/2k)}, we explore a more combinatorial approach. For instance, based on the notion of critical variables, we give an algorithm with running time ${n\choose {\le t}} 2^{n-n/k}$, where $n$ is the number of variables and $t$ is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our algorithm also achieves polynomial space in contrast to the algebraic approach that uses exponential space.

In continuation to our recent work [AJ22] on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results.

THE DEEP CONNECTIONS between logic and automata theory have led to extensive applications in formal specification and verification of systems. In recent years, research has focused on extensibility of such techniques to infinite-state systems. This has led to developing theories of transformations, applications to verification of timed/recursive/concurrent/probabilistic systems, database theory, distributed algorithms, programs running under weak memory models and cryptographic protocols. Logic has also been the ground for foundational research, revisiting classical model theory from computational perspectives. This has led to results in Skolem Lowenheim theorems in the finite, synthesis of Boolean functions and Skolem functions, definability in first-order theories of graphs, algebraic characterizations, block products of algebraic structures, and decidable fragments of first-order modal logics.

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.

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.

The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter.

Let C be an arithmetic circuit of size s, given as input that computes a polynomial $$f\in {\mathbb {F}}[x_1,x_2,\ldots ,x_n]$$ , where $${\mathbb {F}}$$ is a finite field or the field of rationals. Using the Hadamard product of polynomials, we obtain new algorithms for the following two problems first studied by Koutis and Williams (Faster algebraic algorithms for path and packing problems, 2008, https://doi.org/10.1007/978-3-540-70575-8_47 ; ACM Trans Algorithms 12(3):31:1–31:18, 2016, https://doi.org/10.1145/2885499 ; Inf Process Lett 109(6):315–318, 2009, https://doi.org/10.1016/j.ipl.2008.11.004 ): Other results include fast deterministic algorithms for $${{{(\textit{k,n}){-}\mathrm{M{L}\normalsize {C}}}}}$$ and $${{{\textit{k}{-}\mathrm{M{M}\normalsize {D}}}}}$$ problems for depth three circuits.

An arc-colored tournament is said to be k-spanning for an integer k≥ 1 if the union of its arc-color classes of maximal valency at most k is the arc set of a strongly connected digraph. It is proved that isomorphism testing of k-spanning tournaments is fixed-parameter tractable.