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

Syntactic Separation of Subset Satisfiability Problems

Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems

A Size-Free CLT for Poisson Multinomials and its Applications

An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We show that any $(n,k)$-PMD is ${\rm poly}\left({k\over \sigma}\right)$-close in total variation distance to the (appropriately discretized) multi-dimensional Gaussian with the same first two moments, removing the dependence on $n$ from the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is obtained by bootstrapping the Valiant-Valiant CLT itself through the structural characterization of PMDs shown in recent work by Daskalakis, Kamath, and Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS for approximate Nash equilibria in anonymous games, significantly improving the state of the art, and matching qualitatively the running time dependence on $n$ and $1/\varepsilon$ of the best known algorithm for two-strategy anonymous games. Our new CLT also enables the construction of covers for the set of $(n,k)$-PMDs, which are proper and whose size is shown to be essentially optimal. Our cover construction combines our CLT with the Shapley-Folkman theorem and recent sparsification results for Laplacian matrices by Batson, Spielman, and Srivastava. Our cover size lower bound is based on an algebraic geometric construction. Finally, leveraging the structural properties of the Fourier spectrum of PMDs we show that these distributions can be learned from $O_k(1/\varepsilon^2)$ samples in ${\rm poly}_k(1/\varepsilon)$-time, removing the quasi-polynomial dependence of the running time on $1/\varepsilon$ from the algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201

Bi-Criteria and Approximation Algorithms for Restricted Matchings

In this work we study approximation algorithms for the \textit{Bounded Color Matching} problem (a.k.a. Restricted Matching problem) which is defined as follows: given a graph in which each edge $e$ has a color $c_e$ and a profit $p_e \in \mathbb{Q}^+$, we want to compute a maximum (cardinality or profit) matching in which no more than $w_j \in \mathbb{Z}^+$ edges of color $c_j$ are present. This kind of problems, beside the theoretical interest on its own right, emerges in multi-fiber optical networking systems, where we interpret each unique wavelength that can travel through the fiber as a color class and we would like to establish communication between pairs of systems. We study approximation and bi-criteria algorithms for this problem which are based on linear programming techniques and, in particular, on polyhedral characterizations of the natural linear formulation of the problem. In our setting, we allow violations of the bounds $w_j$ and we model our problem as a bi-criteria problem: we have two objectives to optimize namely (a) to maximize the profit (maximum matching) while (b) minimizing the violation of the color bounds. We prove how we can "beat" the integrality gap of the natural linear programming formulation of the problem by allowing only a slight violation of the color bounds. In particular, our main result is \textit{constant} approximation bounds for both criteria of the corresponding bi-criteria optimization problem