Periodic Reporting for period 1 - NoShortProof (Nonexistence of Short Proofs)
Berichtszeitraum: 2024-04-01 bis 2026-03-31
This project investigates the fundamental limits of efficient algorithms for solving combinatorial problems through formal mathematical reasoning frameworks known as proof systems. These systems form the theoretical basis of widely used algorithmic methods such as SAT solvers, algebraic techniques, integer programming, and semidefinite programming hierarchies. Despite their practical success, a theoretical understanding of why these algorithms perform well, or fail, on real-world instances remains larely incomplete. The main objectives are to establish strong impossibility results, specifically average-case lower bounds and supercritical trade-offs, on the size or depth of proofs in three key proof systems: Resolution, Cutting Planes, and Sum-of-Squares. The mathematical methods developed have broader implications, including advances in pseudo-random matrix analysis and circuit complexity. Practically, these results clarify the inherent hardness of fundamental computational and learning problems beyond worst-case scenarios, offering guidance for future algorithm design by ruling out futile heuristics.
The project successfully developed novel techniques addressing open problems in proof complexity. For resolution and cutting planes, a variable compression method was developed to demonstrate, for the first time, robust and supercritical trade-offs between proof width and depth, and between depth and size, relative to input size. These results have implications for monotone circuits and Weisfeiler-Leman algorithms. In the Sum-of-Squares system, foundational work was done for analyzing pseudo-moment matrices related to Non-Gaussian Component Analysis, introduing a new combinatorial concept of graph separators and a novel algebraic approach to proving semi-definiteness via representations of real algebras. The latter shows promise for addressing challenges in the analysis of pseudo-calibration more broadly. Key findings have been published in reputed theoretical computer science conference venues such as FOCS and STOC, with journal versions in preparation. Collaborative interactions with leading experts in computer science and statistics enhanced both methodology and interdisciplinary impact. The project concluded early due to the start of a lectureship at a leading UK institute, with all planned methodologies initiated and several milestones achieved.
The project advanced proof complexity and the theoretical foundations of combinatorial optimization by establishing new complexity trade-offs and average-case lower bounds. Specifically, strong "supercritical trade-off" results were proved for monotone circuits, resolution, cutting planes, and the Weisfeiler-Leman algorithm. These show, for example, that certain functions are computable by small circuits but any such circuit must have depth super-linear or even super-polynomial in the number of variables. A nearly optimal lifting theorem was proved, refining a widely used technique in complexity theory. The first super-constant degree lower bound for Non-Gaussian Component Analysis was established against the Sum-of-Squares hierarchy, significantly advancing prior results on the information-computation gap in statistical learning. A method using representations of real algebras was introduced for analyzing pseudo-random matrices, offering a novel approach to the study of pseudo-calibration.