Greedy Strikes Again: A Deterministic PTAS for Commutative Rank of Matrix Spaces

We consider the problem of commutative rank computation of a given matrix space. A matrix space is a (linear) subspace of the (linear) space of n x n matrices over a given field. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is n, subsumes problems such as testing perfect matching in graphs and identity testing of algebraic branching programs. An efficient deterministic computation of the commutative rank is a major open problem, although there is a simple and efficient randomized algorithm for it. Recently, there has been a series of results on computing the non-commutative rank of matrix spaces in deterministic polynomial time. Since the non-commutative rank of any matrix space is at most twice the commutative rank, one immediately gets a deterministic 1/2-approximation algorithm for the computation of the commutative rank. This leads to a natural question of whether this approximation ratio can be improved. In this paper, we answer this question affirmatively. We present a deterministic Polynomial-time approximation scheme (PTAS) for computing the commutative rank of a given matrix space B. More specifically, given a matrix space and a rational number e > 0, we give an algorithm, that runs in time O(n^(4 + 3/e)) and computes a matrix A in the given matrix space B such that the rank of A is at least (1-e) times the commutative rank of B. The algorithm is the natural greedy algorithm. It always takes the first set of k matrices that will increase the rank of the matrix constructed so far until it does not find any improvement, where the size of the set k depends on e

Variety Membership Testing in Algebraic Complexity Theory

In this thesis, we study some of the central problems in algebraic complexity theory through the lens of the variety membership testing problem. In the first part, we investigate whether separations between algebraic complexity classes can be phrased as instances of the variety membership testing problem. For this, we compare some complexity classes with their closures. We show that monotone commutative single-(source, sink) ABPs are closed. Further, we prove that multi-(source, sink) ABPs are not closed in both the monotone commutative and the noncommutative settings. However, the corresponding complexity classes are closed in all these settings. Next, we observe a separation between the complexity class VQP and the closure of VNP. In the second part, we cover the blackbox polynomial identity testing (PIT) problem, and the rank computation problem of symbolic matrices, both phrasable as instances of the variety membership testing problem. For the blackbox PIT, we give a randomized polynomial time algorithm that uses the number of random bits that matches the information-theoretic lower bound, differing from it only in the lower order terms. For the rank computation problem, we give a deterministic polynomial time approximation scheme (PTAS) when the degrees of the entries of the matrices are bounded by a constant. Finally, we show NP-hardness of two problems on 3-tensors, both of which are instances of the variety membership testing problem. The first problem is the orbit closure containment problem for the action of GLk x GLm x GLn on 3-tensors, while the second problem is to decide whether the slice rank of a given 3-tensor is at most r

On approximate polynomial identity testing and real root finding

In this thesis we study the following three topics, which share a connection through the (arithmetic) circuit complexity of polynomials. 1. Rank of symbolic matrices. 2. Computation of real roots of real sparse polynomials. 3. Complexity of symmetric polynomials. We start with studying the commutative and non-commutative rank of symbolic matrices with linear forms as their entries. Here we show a deterministic polynomial time approximation scheme (PTAS) for computing the commutative rank. Prior to this work, deterministic polynomial time algorithms were known only for computing a 1/2-approximation of the commutative rank. We give two distinct proofs that our algorithm is a PTAS. We also give a min-max characterization of commutative and non-commutative ranks. Thereafter we direct our attention to computation of roots of uni-variate polynomial equations. It is known that solving a system of polynomial equations reduces to solving a uni-variate polynomial equation. We describe a polynomial time algorithm for (n,k,\tau)-nomials which computes approximations of all the real roots (even though it may also compute approximations of some complex roots). Moreover, we also show that the roots of integer trinomials are well-separated. Finally, we study the complexity of symmetric polynomials. It is known that symmetric Boolean functions are easy to compute. In contrast, we show that the assumption VP \neq VNP implies that there exist hard symmetric polynomials. To prove this result, we use an algebraic analogue of the classical Newton iteration.In dieser Dissertation untersuchen wir die folgenden drei Themen, welche durch die (arithmetische) Schaltkreiskomplexität von Polynomen miteinander verbunden sind: 1. der Rang von symbolischen Matrizen, 2. die Berechnung von reellen Nullstellen von dünnbesetzten (“sparse”) Polynomen mit reellen Koeffizienten, 3. die Komplexität von symmetrischen Polynomen. Wir untersuchen zunächst den kommutativen und nicht-kommutativen Rang von Matrizen, deren Einträge aus Linearformen bestehen. Hier beweisen wir die Existenz eines deterministischem Polynomialzeit-Approximationsschemas (PTAS) für die Berechnung des kommutative Ranges. Zuvor waren polynomielle Algorithmen nur für die Berechnung einer 1/2-Approximation des kommutativen Ranges bekannt. Wir geben zwei unterschiedliche Beweise für den Fakt, dass unser Algorithmus tatsächlich ein PTAS ist. Zusätzlich geben wir eine min-max Charakterisierung des kommutativen und nicht-kommutativen Ranges. Anschließend lenken wir unsere Aufmerksamkeit auf die Berechnung von Nullstellen von univariaten polynomiellen Gleichungen. Es ist bekannt, dass das Lösen eines polynomiellem Gleichungssystems auf das Lösen eines univariaten Polynoms zurückgeführt werden kann. Wir geben einen Polynomialzeit-Algorithmus für (n, k, \tau)-Nome, welcher Abschätzungen für alle reellen Nullstellen berechnet (in manchen Fallen auch Abschätzungen von komplexen Nullstellen). Zusätzlich beweisen wir, dass Nullstellen von ganzzahligen Trinomen stets weit voneinander entfernt sind. Schließlich untersuchen wir die Komplexität von symmetrischen Polynomen. Es ist bereits bekannt, dass sich symmetrische Boolesche Funktionen leicht berechnen lassen. Im Gegensatz dazu zeigen wir, dass die Annahme VP \neq VNP bedeutet, dass auch harte symmetrische Polynome existieren. Um dies zu beweisen benutzen wir ein algebraisches Analog zum klassischen Newton-Verfahren

Algebraic and Combinatorial Methods in Computational Complexity

Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science. The seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Quantum Hamiltonian Complexity

Constraint satisfaction problems are a central pillar of modern computational complexity theory. This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum constraint satisfaction problems. Over the past decade and a half, this field has witnessed fundamental breakthroughs, ranging from the establishment of a "Quantum Cook-Levin Theorem" to deep insights into the structure of 1D low-temperature quantum systems via so-called area laws. Our aim here is to provide a computer science-oriented introduction to the subject in order to help bridge the language barrier between computer scientists and physicists in the field. As such, we include the following in this survey: (1) The motivations and history of the field, (2) a glossary of condensed matter physics terms explained in computer-science friendly language, (3) overviews of central ideas from condensed matter physics, such as indistinguishable particles, mean field theory, tensor networks, and area laws, and (4) brief expositions of selected computer science-based results in the area. For example, as part of the latter, we provide a novel information theoretic presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.Comment: v4: published version, 127 pages, introduction expanded to include brief introduction to quantum information, brief list of some recent developments added, minor changes throughou