Algebraic Formula Lower Bounds and Applications

Introduction

Does efficient verification imply efficient search? Can randomness provide massive speed-ups in computation? These are fundamental questions in theoretical computer science, known as P vs. NP and P vs. BPP respectively. Progress on these questions requires us to show that certain computational problems are inherently intractable, i.e., do not admit efficient solutions. An important, and concrete, approach to such questions is to understand the complexity of algebraic problems such as the Determinant and the Permanent, in algebraic models of computation. The aim of this project is to tackle these questions head-on.

Recent Progress

Recent results of the PI and his collaborators have made progress on these problems by resolving a three-decade-old open question. These results show intractability for algebraic models of bounded depth, which is a step towards P vs. NP.

As a consequence, they also imply the first sub-exponential time deterministic algorithms for the important Polynomial Identity Testing (PIT) problem in these settings, which is progress towards P vs. BPP.

Future Goals

However, this barely scratches the surface of what we want to achieve. The aim of this project is to push beyond these state-of-the-art results in many directions. The principal goals include:

1. Proving the first lower bounds for algebraic formulas, which would be a huge breakthrough.
2. Improving our recently obtained lower bounds in order to devise faster PIT algorithms.
3. Showing lower bounds over finite fields.
4. Exploring the applications of such work in related areas such as algebraic proof complexity and algorithm design.

Methodology

The recent results point out a new way of exploiting structural and algebraic techniques to prove lower bounds. The aim is to develop and systematically investigate these techniques, incorporating methods from related fields of mathematics such as the theory of symmetric functions and representation theory, to accomplish these goals.