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

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples

Memristor-based hardware and algorithms for higher-order Hopfield optimization solver outperforming quadratic Ising machines

Ising solvers offer a promising physics-based approach to tackle the challenging class of combinatorial optimization problems. However, typical solvers operate in a quadratic energy space, having only pair-wise coupling elements which already dominate area and energy. We show that such quadratization can cause severe problems: increased dimensionality, a rugged search landscape, and misalignment with the original objective function. Here, we design and quantify a higher-order Hopfield optimization solver, with 28nm CMOS technology and memristive couplings for lower area and energy computations. We combine algorithmic and circuit analysis to show quantitative advantages over quadratic Ising Machines (IM)s, yielding 48x and 72x reduction in time-to-solution (TTS) and energy-to-solution (ETS) respectively for Boolean satisfiability problems of 150 variables, with favorable scaling

Fourier sum of squares certificates

The non-negativity of a function on a finite abelian group can be certified by its Fourier sum of squares (FSOS). In this paper, we propose a method of certifying the non-negativity of an integer-valued function by an FSOS certificate, which is defined to be an FSOS with a small error. We prove the existence of exponentially sparse polynomial and rational FSOS certificates and we provide two methods to validate them. As a consequence of the aforementioned existence theorems, we propose a semidefinite programming (SDP)-based algorithm to efficiently compute a sparse FSOS certificate. For applications, we consider certificate problems for maximum satisfiability (MAX-SAT) and maximum k-colorable subgraph (MkCS) and demonstrate our theoretical results and algorithm by numerical experiments

Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis

This thesis settles a number of questions related to computational complexity and algebraic, semidefinite programming based relaxations in optimization and control.Comment: PhD Thesis, MIT, September, 201

On streaming approximation algorithms for constraint satisfaction problems

In this thesis, we explore streaming algorithms for approximating constraint satisfaction problems (CSPs). The setup is roughly the following: A computer has limited memory space, sees a long "stream" of local constraints on a set of variables, and tries to estimate how many of the constraints may be simultaneously satisfied. The past ten years have seen a number of works in this area, and this thesis includes both expository material and novel contributions. Throughout, we emphasize connections to the broader theories of CSPs, approximability, and streaming models, and highlight interesting open problems. The first part of our thesis is expository: We present aspects of previous works that completely characterize the approximability of specific CSPs like Max-Cut and Max-Dicut with $\sqrt{n}$-space streaming algorithm (on $n$-variable instances), while characterizing the approximability of all CSPs in $\sqrt n$ space in the special case of "composable" (i.e., sketching) algorithms, and of a particular subclass of CSPs with linear-space streaming algorithms. In the second part of the thesis, we present two of our own joint works. We begin with a work with Madhu Sudan and Santhoshini Velusamy in which we prove linear-space streaming approximation-resistance for all ordering CSPs (OCSPs), which are "CSP-like" problems maximizing over sets of permutations. Next, we present joint work with Joanna Boyland, Michael Hwang, Tarun Prasad, and Santhoshini Velusamy in which we investigate the $\sqrt n$-space streaming approximability of symmetric Boolean CSPs with negations. We give explicit $\sqrt n$-space sketching approximability ratios for several families of CSPs, including Max-$k$AND; develop simpler optimal sketching approximation algorithms for threshold predicates; and show that previous lower bounds fail to characterize the $\sqrt n$-space streaming approximability of Max-$3$AND.Comment: Harvard College senior thesis; 119 pages plus references; abstract shortened for arXiv; formatted with Dissertate template (feel free to copy!); exposits papers arXiv:2105.01782 (APPROX 2021) and arXiv:2112.06319 (APPROX 2022

Duality methods in networks, computer science models, and disordered condensed matter systems

In this thesis, I explore lattice independent duality and systems to which it can be applied. I first demonstrate classical duality on models in an external field, including the Ising, Potts, and x-y models, showing in particular how this modifies duality to be lattice independent and applicable to networks. I then present a novel application of duality on the boolean satsifiability problem, one of the most important problems in computational complexity, through mapping to a low temperature Ising model. This establishes the equivalence between boolean satisfiability and a problem of enumerating the positive solutions to a Diophantine system of equations. I continue by combining duality with a prominent tool for models on networks, belief propagation, deriving a new message passing procedure, dual belief propagation. In the final part of my thesis, I shift to propose and examine a semiclassical model, the two-component Coulomb glass model, which can explain the giant magnetoresistance peak present in disordered films near a superconductor-insulator transition as the effect of competition between single particle and localized pair transport. I numerically analyze the density of states and transport properties of this model

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers