CAREER: Efficient Fine-grained Algorithms

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Ever since the inception of computation, the fundamental question has been- what problems can be solved by computers? And which ones can be solved in a reasonable time? This brought in the notion of polynomial time solvable problems which are considered efficient vs NP-hard problems which take prohibitively long time to solve. The field of approximation algorithms, along with the theory of inapproximability, deals with which of the NP-hard problems can be solved efficiently by allowing approximate solutions. However, this crude distinction of algorithmic efficiency--polynomial vs NP-hard, is insufficient when handling today's large scale of data. We need a finer-grained design and analysis of algorithms that pinpoints the exact exponent of polynomial running time, and a better understanding of when a speed-up is not possible. Unfortunately, except for a few problem-specific innovations, the study of such algorithms is deeply lacking in the literature.This project targets to build a unified theory of fine-grained algorithm design to study fast approximation algorithms, and their fine-grained hardness. Developing systematic techniques that emphasize on the trade-offs between running time, approximation and randomness, and aid in designing low-complexity parallel algorithms will significantly improve the state of the art. Moreover, motivated by core machine learning applications, the project proposes an alternate model of efficiency via classical query complexity. Tools from classical query complexity have previously inspired development of fast algorithms and vice versa; the PI expects similar connections to happen in this project. Elements of this endeavor will be integrated with new courses, many of the algorithms developed herein will be implemented and the close connection of the PI with industry will result in possible adaptation of methodologies.The project will address a suite of important optimization problems, from long-standing open questions to modern problems with diverse applications. It will introduce fresh tools from additive combinatorics, fourier analysis, rate distortion theory, and circuit complexity for analysis of algorithms, and establishing lower bounds. Specifically, new generic techniques of amnesic dynamic programming, fast matrix-product over semiring, and low-degree polynomial method will be developed to design fine-grained approximation algorithms and their parallel counterparts. Time complexity may not always be the primary measure of efficiency. There are many core machine learning problems where query complexity, that quantifies the amount of labeled data acquired via active querying, is more important. The project will analyze for the first time the query complexity of basic learning problems, and explore its connections to developing fast algorithms.