Abstract We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (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 Fürer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison with 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.

In this note we explore the color refinement procedure — also known as the 1-dimensional Weisfeiler-Leman procedure, well-studied in connection with the Graph Isomorphism problem — in the context of the Graph Reconstruction conjecture of Ulam. A basic fact about the Ulam reconstruction conjecture is that the connectedness of a graph is determined by the deck of its vertex-deleted subgraphs, which are considered up to isomorphism. We strengthen this result by proving that connectedness of a graph can even be determined from the deck of its vertex-deleted subgraphs given only by their stable colorings (i.e., up to equivalence under color refinement). It follows as a consequence that connectedness is recognizable by Reconstruction Graph Neural Networks, which is a recently introduced GNN architecture inspired by the reconstruction conjecture (Cotta, Morris, Ribeiro 2021).

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$x^{\prime}\in X_{j}$. 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$\mathbb{Q}\langle X\rangle$[1]. 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 [2]–[4]. En-route, we obtain the first deterministic polynomial-time algorithm for the equivalence testing problem of$k$-tape weighted automata (for constant$k$) resolving a longstanding open problem [5], [6]. Algebraically, the equivalence problem reduces to testing whether a partially commutative rational series over the partitioned set$X$is zero or not [6]. Decidability of this problem was established by Harju and Karhumäki [5]. Prior to this work, a randomized polynomial-time algorithm for this problem was given by Worrell [6] and, subsequently, a deterministic quasipolynomial-time algorithm was also developed [7].

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 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 \cite{Co90,Cohnfir} 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 $\RIT$ 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.

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (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 Fürer'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.

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.

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, 1. Given an n -variate noncommutative polynomial f ∈ F 〈 X 〉 over a field F as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial g ∈ F 〈 x , y 〉 ; the reduction transforms f into a circuit for g , and given a complete factorization of g , the reduction recovers a complete factorization of f in polynomial time. The reduction works both in the white-box and the black-box setting. 2. We show over the field of rationals that bivariate linear matrix factorization problem for 4 × 4 matrices is at least as hard as factoring square-free integers and for 3 × 3 matrices it is in polynomial time.

Hrubeš and Wigderson [Hrubeš 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, there are deterministic polynomial-time algorithms due to Garg, Gurvits, Oliveira, and Wigderson [Ankit Garg et al., 2016] and Ivanyos, Qiao, and Subrahmanyam [Ivanyos et al., 2018]. A central open problem in this area is to design an efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this 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 concepts from matrix coefficient realization theory [Volčič, 2018] and properties of cyclic division algebras [T.Y. Lam, 2001]. En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas [Michael A. Forbes and Amir Shpilka, 2013] inside a cyclic division algebra of small index.

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 $${\mathbb {F}}_2$$ each of which is a product of affine forms. We focus on the case of k-CNF formulas (the $${k}-\textsc {Sub}-\textsc {Sat}$$ problem). Clearly, $${k}-\textsc {Sub}-\textsc {Sat}$$ is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that $${2}-\textsc {Sub}-\textsc {Sat}$$ is NP-hard, and $${\small \mathrm {W}}[1]$$ -hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max- $${2}-\textsc {Sub}-\textsc {Sat}$$ 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 running time $$O^*(1.5)^r$$ for $${2}-\textsc {Sub}-\textsc {Sat}$$ , where r is the subspace dimension, as well as an $$O^*(1.4312)^n$$ time algorithm where n is the number of variables. Turning to $${k}-\textsc {Sub}-\textsc {Sat}$$ for $$k \geqslant 3$$ , while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time $$\approx 2^{r(1-1/2k)}$$ , we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized k-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with running time $$\approx {n\atopwithdelims (){\leqslant t}} 2^{n-n/k}$$ 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 combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for $${k}-\textsc {Sub}-\textsc {Sat}$$ with running time $$\approx 2^{n-n/2k}$$ . This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of O(n) polynomial equations in n variables over $${\mathbb {F}}_2$$ , we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick.

Let \({\mathbb {F}}[X]\) be the polynomial ring in the variables X = { x 1 , x 2 ,…, x n } over a field \({\mathbb {F}}\) . An ideal I = 〈 p 1 ( x 1 ),…, p n ( x n )〉 generated by univariate polynomials \(\{p_{i}(x_{i})\}_{i=1}^{n}\) is a univariate ideal . Motivated by Alon’s Combinatorial Nullstellensatz we study the complexity of univariate ideal membership : Given \(f\in {\mathbb {F}}[X]\) by a circuit and polynomials p i the problem is test if f ∈ I . We obtain the following results. Suppose f is a degree- d , rank- r polynomial given by an arithmetic circuit where ℓ i : 1 ≤ i ≤ r are linear forms in X . We give a deterministic time d O ( r ) ⋅poly( n ) division algorithm for evaluating the (unique) remainder polynomial f ( X )mod I at any point \(\vec {a}\in {\mathbb {F}}^{n}\) . This yields a randomized n O ( r ) algorithm for minimum vertex cover in graphs with rank- r adjacency matrices. It also yields a new n O ( r ) algorithm for evaluating the permanent of a n × n matrix of rank r , over any field \(\mathbb {F}\) . Let f be over rationals with \(\deg (f)=k\) treated as fixed parameter. When the ideal \(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{n}^{e_{n}}}\right \rangle \) , we can test ideal membership in randomized O ∗ ((2 e ) k ). On the other hand, if each p i has all distinct rational roots we can check if f ∈ I in randomized O ∗ ( n k /2 ) time, improving on the brute-force \(\left (\begin {array}{cc}{n+k}\\ k \end {array}\right )\) -time search. If \(I=\left \langle {p_{1}(x_{1}), \ldots , p_{k}(x_{k})}\right \rangle \) , with k as fixed parameter, then ideal membership testing is W[2]-hard. The problem is MINI[1]-hard in the special case when \(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{k}^{e_{k}}}\right \rangle \) .

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.

We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Brešar-Klep theorem [Brešar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q over rationals. 1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q<(X)>. Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem. 2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f. 3) The Brešar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d. Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].

Hrubeš and Wigderson [Pavel Hrubeš 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.

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.

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.

Let $G =\langle S\rangle$ be a solvable permutation group given as input by the generating set $S$, that is, $G$ is a solvable subgroup of the symmetric group $S_n$. We give a deterministic polynomial-time algorithm that computes an expanding generating set $T$ of size $\tilde{O}(n^2(1/\lambda)^{c})$ for $G$ such that the undirected Cayley graph ${\rm Cay}_u(G,T)$ is a $\lambda$-spectral expander and the constant $c$ is at most $8$ (the $\tilde{O}$ notation suppresses $\log ^{O(1)}n$ and $\log ^{O(1)}(1/\lambda)$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n (\log d)^{O(1)}(1/\varepsilon)^{c})$ for the groups $\mathbb{Z}_d^n$ and $c\leq 8$. The earlier known size bound was $O((d + n/\varepsilon^2)^{11/2})$ given by [ Y. Azar, R. Motwani, and J. Naor , Combinatorica, 18 (1998), pp. 151--171]. We also note that for any permutation group $G\le S_n$ given by a generating set, in deterministic polynomial time we can compute an expanding generating set $T$ of size $\left({n}/{\lambda}\right)^{O(1)}$ such that ${\rm Cay}_u(G,T)$ is a $\lambda$-spectral expander where the $O(1)$ notation involves a large constant.

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 \pi from X_1 to X_2 such that i^\pi\ne 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 even if a conjunction of disjunctions of such constraints is allowed. As a consequence of the main result 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 number of moved vertices is 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 \pi 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. In the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most k vertices, shown to be in FPT by Schweitzer [ESA 2011].

We show that checking if a given hypergraph has an automorphism that moves exactly k vertices is fixed parameter tractable, using k and additionally either the maximum hyperedge size or the maximum color class size as parameters. In particular, it suffices to use k as parameter if the hyperedge size is at most polylogarithmic in the size of the given hypergraph. As a building block for our algorithms, we generalize Schweitzer's FPT algorithm [ESA 2011] that, given two graphs on the same vertex set and a parameter k, decides whether there is an isomorphism between the two graphs that moves at most k vertices. We extend this result to hypergraphs, using the maximum hyperedge size as a second parameter. Another key component of our algorithm is an orbit-shrinking technique that preserves permutations that move few points and that may be of independent interest. Applying it to a suitable subgroup of the automorphism group allows us to switch from bounded hyperedge size to bounded color classes in the exactly-k case.

In this paper we show that black-box polynomial identity testing for noncommutative polynomials f∈𝔽⟨z1,z2,…,zn⟩ of degree D and sparsity t, can be done in randomized (n,logt,logD) time. As a consequence, given a circuit C of size s computing a polynomial f∈𝔽⟨ z1,z2,…,zn⟩ with at most t non-zero monomials, then testing if f is identically zero can be done by a randomized algorithm with running time polynomial in s and n and logt. This makes significant progress on a question that has been open for over ten years. Our algorithm is based on automata-theoretic ideas that can efficiently isolate a monomial in the given polynomial. In particular, we carry out the monomial isolation using nondeterministic automata.

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.

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism, denoted CHI, which has running time (2 b N) O(1), where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe an fpt algorithm for a parameterized coset intersection problem that is used as a subroutine in our algorithm for CHI.

We give the first polynomial-time algorithm for checking whether the Galois group Gal( f ) of an input polynomial f ( X ) ∈ Q[ X ] is nilpotent: the running time of our algorithm is bounded by a polynomial in the size of the coefficients of f and the degree of f . Additionally, we give a deterministic polynomial-time algorithm that, when given as input a polynomial f ( X ) ∈ Q[ X ] with nilpotent Galois group, computes for each prime factor p of # Gal( f ), a polynomial g p ( X )∈ Q[ X ] whose Galois group of is the p -Sylow subgroup of Gal( f ).

Using $\varepsilon$-bias spaces over $\F_2$, we show that the Remote Point Problem (RPP), introduced by Alon et al \cite{APY09}, has an $\NC^2$ algorithm (achieving the same parameters as \cite{APY09}). We study a generalization of the Remote Point Problem to groups: we replace $\F_2^n$ by $\mcG^n$ for an arbitrary fixed group $\mcG$. When $\mcG$ is Abelian we give an $\NC^2$ algorithm for RPP, again using $\varepsilon$-bias spaces. For nonabelian $\mcG$, we give a deterministic polynomial-time algorithm for RPP. We also show the connection to construction of expanding generator sets for the group $\mcG^n$. All our algorithms for the RPP achieve essentially the same parameters as \cite{APY09}.

\begin{abstract}

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. \begin{itemize} \item[$\bullet$] 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 $\ModpL/\Poly$. \item[$\bullet$] 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. \end{itemize}

Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum algorithm with query complexity sub-linear in the number of generators for the ring, when the number of indeterminates of the input polynomial is small (ideally a constant). Towards a lower bound, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem.

Using ideas from automata theory we design a new efficient (deterministic) identity test for the noncommutative polynomial identity testing problem (first introduced and studied in [RS05, BW05]). More precisely, given as input a noncommutative circuit C{x1, ldrldrldr , xn} computing a polynomial in F{x1, ldrldrldr , xn} of degree d with at most t monomials, where the variables xiare noncommuting, we give a deterministic polynomial identity test that checks if C equiv 0 and runs in time polynomial in d, n, |C|, and t. The same methods works in a black-box setting: given a noncommuting black-box polynomial f isin F{x1, ldrldrldr , xn} of degree d with t monomials we can, in fact, reconstruct the entire polynomial f in time polynomial in n, d and t. Indeed, we apply this idea to the reconstruction of black-box noncommuting algebraic branching programs (the ABPs considered by Nisan in [N91] and Raz-Shpilka in [RS05]). Assuming that the black-box model allows us to query the ABP for the output at any given gate then we can reconstruct an (equivalent) ABP in deterministic polynomial time. Finally, we turn to commutative identity testing and explore the complexity of the problem when the coefficients of the input polynomial come from an arbitrary finite commutative ring with unity whose elements are uniformly encoded as strings and the ring operations are given by an oracle. We show that several algorithmic results for polynomial identity testing over fields also hold when the coefficients come from such finite rings.

We study the algorithmic complexity of lattice problems based on the sieving technique due to Ajtai, Kumar, and Sivakumar~\cite{aks}. Given a $k$-dimensional subspace $M\subseteq \R^n$ and a full rank integer lattice $\L\subseteq \Q^n$, the \emph{subspace avoiding problem} SAP, defined by Bl\"omer and Naewe \cite{blomer}, is to find a shortest vector in $\L\setminus M$. We first give a $2^{O(n+k \log k)}$ time algorithm to solve \emph{the subspace avoiding problem}. Applying this algorithm we obtain the following results. \begin{enumerate} \item We give a $2^{O(n)}$ time algorithm to compute $i^{th}$ successive minima of a full rank lattice $\L\subset \Q^n$ if $i$ is $O(\frac{n}{\log n})$. \item We give a $2^{O(n)}$ time algorithm to solve a restricted \emph{closest vector problem CVP} where the inputs fulfil a promise about the distance of the input vector from the lattice. \item We also show that unrestricted CVP has a $2^{O(n)}$ exact algorithm if there is a $2^{O(n)}$ time exact algorithm for solving CVP with additional input $v_i\in \L, 1\leq i\leq n$, where $\|v_i\|_p$ is the $i^{th}$ successive minima of $\L$ for each $i$. \end{enumerate} We also give a new approximation algorithm for SAP and the \emph{Convex Body Avoiding problem} which is a generalization of SAP. Several of our algorithms work for \emph{gauge} functions as metric, where the gauge function has a natural restriction and is accessed by an oracle.

A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, $D(G)$, is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph.

In this paper we study the complexity of bounded color multiplicity graph isomorphism BCGI/sub b/: the input is a pair of vertex-colored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGI/sub b/ is in the #L hierarchy (more precisely, the Mod/sub k/L hierarchy for some constant k depending on b). Combined with the fact that bounded color multiplicity graph isomorphism is logspace many-one hard for every set in the Mod/sub k/L hierarchy for any constant k, we get a tight classification of the problem using logspace-bounded counting classes.

The group isomorphism problem consists in deciding whether two input groups G/sup 1/ and G/sup 2/ given by their multiplication tables are isomorphic. We first give a 2-round Arthur-Merlin protocol for the group non-isomorphism problem such that on input groups (G/sup 1/, G/sup 2/) of size n, Arthur uses O(log/sup 6/ n) random bits and Merlin uses O(log/sup 2/ n) nondeterministic bits. We derandomize this protocol for the case of solvable groups showing the following two results: (a) We give a uniform NP machine for solvable group non-isomorphism, that works correctly on all but 2/sup polylog(n)/ inputs of any length n. Furthermore, this NP machine is always correct when the input groups are nonisomorphic. The NP machine is obtained by an unconditional derandomization of the AM protocol. (b) Under the assumption that EXP /spl nsube/ i.o.PSPACE we get a complete derandomization of the above AM protocol. Thus, EXP /spl nsube/ i.o.PSPACE implies that group isomorphism for solvable groups is in NP /spl cap/ coNP.

The goal of this paper is to classify abelian permutation group problems using logspace counting classes. Building on McKenzie and Cook's [MC87] classification of permutation group problems into four NC Turing-equivalent sets, we show that all these problems are essentially captured by the generalized logspace mod-class ModL, where ModL is the logspace analogue of ModP (defined by Kobler and Toda (KT96)). More precisely, our results are as follows: 1. For abelian permutation groups, the problems of membership testing, isomorphism testing and computing the order of a group are all in ZPL/sup ModL/, and are all hard for ModL under logspace Turing reductions. 2. The problems of computing the intersection of abelian permutation groups, and computing a generator-relator presentation for a given abelian permutation group are in FL/sup ModL//poly. Furthermore, the search version of membership testing is also in FL/sup ModL//poly.

We show that graph isomorphism is in the complexity class SPP and hence it is in /spl oplus/P (in fact, it is in Mod/sub k/P for each k/spl ges/2). We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FP SPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem for permutation groups, studied in the context of quantum algorithms, has an FP/sup SPP/ algorithm. Also, some other algorithmic problems over permutation groups known to be at least as hard as graph isomorphism (e.g. coset intersection) are in SPP, and thus in Mod/sub k/P for each k>2.

Motivated by the question of the relative complexities of the graph isomorphism and the graph automorphism problems, we define and study the modular graph automorphism problems. These are the decision problems mod k -GA which consist, for each k > 1, of deciding whether the number of automorphisms of a graph is divisible by k. The mod k -GA problems all turn out to be intermediate in difficulty between graph automorphism and graph isomorphism. We define an appropriate search problem corresponding to mod k -GA and design an algorithm that polynomial-time reduces the mod k -GA search problem to the decision problem. Combining this algorithm with an IP protocol, we obtain a randomized polynomial-time checker for mod$_{k}$-GA $\forall k>1$.

In this paper we design a nonadaptive NC checker for permutation group intersection, sharpening a result from M. Blum and S. Kannan (1995). This is a consequence of two results. First we show that a nontrivial permutation in the intersection of two given permutation groups (described by lists of generators) can be computed by an NC algorithm with one round of parallel queries to the group intersection problem. Next we design a two-round interactive proof system for the complement of the group intersection problem, for which the honest prover can be simulated by an NC algorithm with one round of parallel queries to group intersection. As a consequence we also have nonadaptive NC checkers for some related group-theoretic problems. On the technical side, we define a generalization of wreath products of permutation groups. This product plays a crucial role in the design of the nonadaptive checkers.

We show that for any graph G, k non-trivial automorphisms of G-if as many exist-can be computed in time |G|/sup O(log k/) with nonadaptive queries to GA, the decision problem for Graph Automorphism. As a consequence we show that some problems related to GA and GI are polynomial-time truth-table equivalent to GA.

The problem of optimization in a multiple-cost search space by combining admissible heuristic estimates for the different cost parameters is investigated. The authors propose an algorithm, MULT* for solving trade-off optimization problems for which there are good admissible heuristics available for each of the associated cost parameters. It is shown that MULT* is an admissible algorithm and it has many of the important properties of A*. Conditions are also given under which it is possible to prune paths in the search graph. A method is also given to relax admissibility of the heuristics to have a more efficient version of MULT* with a bounded decrease in solution quality.>