SubsidieMeestersAlgebraic Formula Lower Bounds and Applications
This project aims to establish lower bounds for algebraic formulas and improve Polynomial Identity Testing algorithms by leveraging structural and algebraic techniques in theoretical computer science.
Projectdetails
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:
- Proving the first lower bounds for algebraic formulas, which would be a huge breakthrough.
- Improving our recently obtained lower bounds in order to devise faster PIT algorithms.
- Showing lower bounds over finite fields.
- 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.
Financiële details & Tijdlijn
Financiële details
| Subsidiebedrag | € 1.869.055 |
| Totale projectbegroting | € 1.869.055 |
Tijdlijn
| Startdatum | 1-6-2024 |
| Einddatum | 31-5-2029 |
| Subsidiejaar | 2024 |
Partners & Locaties
Projectpartners
- KOBENHAVNS UNIVERSITETpenvoerder
Land(en)
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This project aims to tackle fundamental challenges in polynomial computation and manipulation, seeking breakthroughs in complexity, algorithms, and quantum information theory.
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Develop new computational methods in nonlinear algebra using algebraic geometry to enhance the precision and reliability of numerical integration and algebraic invariant computation.
The Hardness of Finding Good Algorithms
The project investigates the difficulty of finding efficient algorithms for various computational problems, aiming to establish lower bounds and connections to cryptography and learning theory.
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This project aims to develop novel mathematical theories in applied algebra to enhance the analysis of biochemical reaction networks through parametrized polynomial equations.
Polynomial-time Computation: Opening the Blackboxes in Constraint Problems
The project aims to unify and generalize the understanding of the complexity class P through finite-domain constraint satisfaction problems, enhancing insights into efficient computation and its applications.