In this work, we study the computational complexity of quantum determinants, a $q$-deformation of matrix permanents: Given a complex number $q$ on the unit circle in the complex plane and an $n\times n$ matrix $X$, the $q$-permanent of $X$ is defined as $$\mathrm{Per}_q(X) = \sum_{σ\in S_n} q^{\ell(σ)}X_{1,σ(1)}\ldots X_{n,σ(n)},$$ where $\ell(σ)$ is the inversion number of permutation $σ$ in the symmetric group $S_n$ on $n$ elements. The function family generalizes determinant and permanent, which correspond to the cases $q=-1$ and $q=1$ respectively.
For worst-case hardness, by Liouville's approximation theorem and facts from algebraic number theory, we show that for primitive $m$-th root of unity $q$ for odd prime power $m=p^k$, exactly computing $q$-permanent is $\mathsf{Mod}_p\mathsf{P}$-hard. This implies that an efficient algorithm for computing $q$-permanent results in a collapse of the polynomial hierarchy. Next, we show that computing $q$-permanent can be achieved using an oracle that approximates to within a polynomial multiplicative error and a membership oracle for a finite set of algebraic integers. From this, an efficient approximation algorithm would also imply a collapse of the polynomial hierarchy. By random self-reducibility, computing $q$-permanent remains to be hard for a wide range of distributions satisfying a property called the strong autocorrelation property. Specifically, this is proved via a reduction from $1$-permanent to $q$-permanent for $O(1/n^2)$ points $z$ on the unit circle. Since the family of permanent functions shares common algebraic structure, various techniques developed for the hardness of permanent can be generalized to $q$-permanents.
We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite graphs. Furthermore, we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros, ones, and variables. We show that these are exactly the sequences in the complexity class of constant free polynomially sized (weakly) skew circuits.
We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing permutations by signed permutations, and the other by replacing permutations by surjective functions with preimages of prescribed sizes.
The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the determinantal complexity of the d by d permanent polynomial is d^2/2, due to Mignon and Ressayre in 2004. Inspired by their proof method, we introduce a natural rank concept of polynomials, called the bi-polynomial rank. The bi-polynomial rank is related to width of an arithmetic branching program. The bi-polynomial rank of a homogeneous polynomial p of even degree…
The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P = Det(M)$. We prove that the determinantal complexity of the polynomial $\sum_{i = 1}^n x_i^n$ is at least $1.5n - 3$. For every $n$-variate polynomial of degree $d$, the determinantal complexity is trivially at least $d$, and it is a long standing open problem to prove a lower bound which is super linear in $\max\{n,d\}$. Our result is the first lower bound for a…
The fermionant can be seen as a generalization of both the permanent (for $k=-1$) and the determinant. We demonstrate that it is VNP-complete for most cases. Furthermore it is #P-complete for the cases. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (when the Young diagram is a column). We demonstrate that the immanant of any family of Young diagrams with bounded width and at least n boxes at the right of the first column is VNP-complete.
We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation.
Every square matrix $A=(a_{uv})\in \mathcal{C}^{n\times n}$ can be represented as a digraph having $n$ vertices. In the digraph, a block (or 2-connected component) is a maximally connected subdigraph that has no cut-vertex. The determinant and the permanent of a matrix can be calculated in terms of the determinant and the permanent of some specific induced subdigraphs of the blocks in the digraph. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Such partitions of the digraph are called the $\mathcal{B}$-partitions. In this paper, first, we develop a…
We give an introduction to some of the recent ideas that go under the name "geometric complexity theory". We first sketch the proof of the known upper and lower bounds for the determinantal complexity of the permanent. We then introduce the concept of a representation theoretic obstruction, which has close links to algebraic combinatorics, and we explain some of the insights gained so far. In particular, we address very recent insights on the complexity of testing the positivity of Kronecker coefficients. We also briefly discuss the related asymptotic version of this question.
Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show no-go results for the method of shifted partial deriv…
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 ari…
Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to prove VP not equal to VNP, it is sufficient to show that an explicit polynomial in VNP of degree n requires 2^{ω(\sqrt{n}\log n)} size depth-4 circuits. Soon after Tavenas's result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2^{Ω(\sqrt{n}\log n)} have been proved Kayal et al. and Fournier et al. In particular, using combinatorial design Kayal et al.\ construct an explicit polynomial in VNP that requires d…
We show that any nonzero polynomial in the ideal generated by the $r \times r$ minors of an $n \times n$ matrix $X$ can be used to efficiently approximate the determinant. For any nonzero polynomial $f$ in this ideal, we construct a small depth-three $f$-oracle circuit that approximates the determinant of size $Θ(r^{1/3})$ in the sense of border complexity. For many classes of algebraic circuits, this implies that every nonzero polynomial in the ideal generated by $r \times r$ minors is at least as hard to approximately compute as the determinant of size $Θ(r^{1/3})$. We also prove an analogou…
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 $O^{*}({n\choose {\d…
The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_{ws} and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_{n^2}(C), which occur in one coordinate ring of the orbi…
In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes,Wigderson and Yehudayoff [HWY10]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, the…
Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Sta99], in Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84, Sta99]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [HRTS00, OW15] and Geometric complexity theory [IP17]. However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmet…
We introduce and study the notion of read-$k$ projections of the determinant: a polynomial $f \in \mathbb{F}[x_1, \ldots, x_n]$ is called a {\it read-$k$ projection of determinant} if $f=det(M)$, where entries of matrix $M$ are either field elements or variables such that each variable appears at most $k$ times in $M$. A monomial set $S$ is said to be expressible as read-$k$ projection of determinant if there is a read-$k$ projection of determinant $f$ such that the monomial set of $f$ is equal to $S$. We obtain basic results relating read-$k$ determinantal projections to the well-studied noti…
I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.
Equivalence testing for a polynomial family {g_m} over a field F is the following problem: Given black-box access to an n-variate polynomial f(x), where n is the number of variables in g_m, check if there exists an A in GL(n,F) such that f(x) = g_m(Ax). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the determinant (Det) and the two popular variants of the iterated matrix multiplication polynomial: IMM_{w,d} (the (1,1) entry of the product of d many w $\times$ w symbolic matrices) and Tr-IMM_{w,d} (…
The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant. First, we study the complexity of testing the multivariate resultant for zero. Our main result is that this problem is NP-hard under deterministic reductions in any characteristic, for systems of l…
Kaltofen has proposed a new approach in 1992 for computing matrix determinants without divisions. The algorithm is based on a baby steps/giant steps construction of Krylov subspaces, and computes the determinant as the constant term of a characteristic polynomial. For matrices over an abstract ring, by the results of Baur and Strassen, the determinant algorithm, actually a straight-line program, leads to an algorithm with the same complexity for computing the adjoint of a matrix. However, the latter adjoint algorithm is obtained by the reverse mode of automatic differentiation, hence somehow i…
In (Kabanets, Impagliazzo, 2004) it is shown how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family for arithmetical circuits. In this paper, a special case of CPIT is considered, namely low-degree non-singular matrix completion (NSMC). For this subclass of problems it is shown how to obtain the same deterministic time bound, using a weaker assumption in terms of determinantal complexity. Hardness-randomness tradeoffs will also be shown in the converse direction, in an effort to…
Motivated by the recent result of [EJT10] showing that MSO properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and prove that it is L-complete. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicate. We extend this technique to count the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs, and further to counting the number of spanning trees and the number of E…
In this paper, we study the complexity of computing the determinant of a matrix over a non-commutative algebra. In particular, we ask the question, "over which algebras, is the determinant easier to compute than the permanent?" Towards resolving this question, we show the following hardness and easiness of noncommutative determinant computation. * [Hardness] Computing the determinant of an n \times n matrix whose entries are themselves 2 \times 2 matrices over a field is as hard as computing the permanent over the field. This extends the recent result of Arvind and Srinivasan, who proved a sim…
Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output. Wiede-mann's algorithm projects the Krylov sequence obtained by repeatedly multiplying a vector by a matrix to obtain a linearly recurrent sequence. The minimal polynomial of this sequence divides the minimal polynomial of the matrix. For instance, if the $n\times n$ input matrix is sparse with n 1+o(1) non-zero entries, the computation of the sequence is quadratic in the dimension of the matr…
We discuss the connection between computational social choice (comsoc) and computational complexity. We stress the work so far on, and urge continued focus on, two less-recognized aspects of this connection. Firstly, this is very much a two-way street: Everyone knows complexity classification is used in comsoc, but we also highlight benefits to complexity that have arisen from its use in comsoc. Secondly, more subtle, less-known complexity tools often can be very productively used in comsoc.
