Periodic Reporting for period 4 - PCPABF (Challenging Computational Infeasibility: PCP and Boolean functions)
Berichtszeitraum: 2024-04-01 bis 2025-09-30
The project sits in computational complexity and the analysis of algorithms, focusing on when algorithmic problems are provably infeasible to solve or even to approximate efficiently. Classic results show that many NP-hard optimization problems can’t be approximated efficiently unless P = NP. The PCPABF pushes this frontier further by defining the following objectives:
• Objective: advance Khot’s Unique Games Conjecture (UGC) using probabilistically checkable proofs (PCPs) and the analysis of Boolean functions.
Technical focus: study expansion and structure in high-dimensional combinatorial objects (like Grassmann and Johnson graphs) and develop new tools in Boolean function analysis, sharp thresholds, and hardness of approximation.
• Objective: analyzing computational problems via linear-algebraic or lattice-based frameworks.
Technical focus: Formalizing computational problems in algebraic rather than purely combinatorial ones. Reducing diverse computational or decision/optimization problems to standard lattice problems (SVP, CVP, lattice-basis problems, integer-kernel/image problems, etc.) or to problems in integer linear algebra. Studying the complexity (hardness, approximability, algorithmic solvability) of these lattice / linear-algebra derived problems, often in high dimension, and understanding how hardness scales with dimension, approximation factor, norm choice, and basis quality. Exploring what this algebraic/lattice viewpoint reveals about the nature of computational hardness, potentially providing evidence for hardness beyond NP (e.g. showing “intermediate complexity”), or building cryptographic primitives (or proofs) based on that hardness. Developing or harnessing algebraic / geometric methods (basis reduction, normal forms, integer matrix theory, dual lattices / duality, geometry-of-numbers, lattice-based reductions) to tackle problems that traditional combinatorial methods struggle with.
• Objective: advance Khot’s Unique Games Conjecture (UGC) using probabilistically checkable proofs (PCPs) and the analysis of Boolean functions.
Technical focus: study expansion and structure in high-dimensional combinatorial objects (like Grassmann and Johnson graphs) and develop new tools in Boolean function analysis, sharp thresholds, and hardness of approximation.
• Objective: analyzing computational problems via linear-algebraic or lattice-based frameworks.
Technical focus: Formalizing computational problems in algebraic rather than purely combinatorial ones. Reducing diverse computational or decision/optimization problems to standard lattice problems (SVP, CVP, lattice-basis problems, integer-kernel/image problems, etc.) or to problems in integer linear algebra. Studying the complexity (hardness, approximability, algorithmic solvability) of these lattice / linear-algebra derived problems, often in high dimension, and understanding how hardness scales with dimension, approximation factor, norm choice, and basis quality. Exploring what this algebraic/lattice viewpoint reveals about the nature of computational hardness, potentially providing evidence for hardness beyond NP (e.g. showing “intermediate complexity”), or building cryptographic primitives (or proofs) based on that hardness. Developing or harnessing algebraic / geometric methods (basis reduction, normal forms, integer matrix theory, dual lattices / duality, geometry-of-numbers, lattice-based reductions) to tackle problems that traditional combinatorial methods struggle with.
Within the framework of the PCPABF project the team focused on the following scientific goals:
• Analysis of Boolean functions
We developed new analytic tools for studying Boolean functions with small total influence, in collaboration with G. Kindler and D. Minzer. Our results include new proofs of classical theorems by Kahn–Kalai–Linial, Friedgut, and Talagrand, replacing hypercontractivity with a unified approach based on random restrictions and the log-Sobolev inequality. We also made substantial progress toward the Fourier-Entropy Influence Conjecture by establishing improved bounds on the concentration of the Fourier spectrum and extending prior work of Bourgain–Kalai.
• NP-hardness of Almost Coloring Almost 3-Colorable Graphs
To advance hardness results for graph coloring, we studied relaxed “almost-coloring’’ settings. We proved that even when a graph is almost 3-colorable and we may ignore a vanishing fraction of vertices, coloring the remainder with any constant number of colors remains NP-hard. This narrows the possibility for efficient algorithms for 3-colorable graphs.
• Mathematics of Computation – Lattice-Based Framework
Responding to rapid development in lattice-based complexity theory, we produced a monograph outlining a new perspective on computational complexity through linear equations and lattices, including techniques, recent progress, and open problems. This work connects to an invited ICM 2022 lecture and was published by EMS Press in 2023.
• Computational Problems Over Lattices
We investigated structural barriers in lattice algorithms, specifically the difficulty of finding a good basis. We fully characterized non-standard lattices—those lacking a basis that achieves the successive minima—clarifying when shortest vectors fail to form an optimal basis.
• Analysis of Boolean functions
We developed new analytic tools for studying Boolean functions with small total influence, in collaboration with G. Kindler and D. Minzer. Our results include new proofs of classical theorems by Kahn–Kalai–Linial, Friedgut, and Talagrand, replacing hypercontractivity with a unified approach based on random restrictions and the log-Sobolev inequality. We also made substantial progress toward the Fourier-Entropy Influence Conjecture by establishing improved bounds on the concentration of the Fourier spectrum and extending prior work of Bourgain–Kalai.
• NP-hardness of Almost Coloring Almost 3-Colorable Graphs
To advance hardness results for graph coloring, we studied relaxed “almost-coloring’’ settings. We proved that even when a graph is almost 3-colorable and we may ignore a vanishing fraction of vertices, coloring the remainder with any constant number of colors remains NP-hard. This narrows the possibility for efficient algorithms for 3-colorable graphs.
• Mathematics of Computation – Lattice-Based Framework
Responding to rapid development in lattice-based complexity theory, we produced a monograph outlining a new perspective on computational complexity through linear equations and lattices, including techniques, recent progress, and open problems. This work connects to an invited ICM 2022 lecture and was published by EMS Press in 2023.
• Computational Problems Over Lattices
We investigated structural barriers in lattice algorithms, specifically the difficulty of finding a good basis. We fully characterized non-standard lattices—those lacking a basis that achieves the successive minima—clarifying when shortest vectors fail to form an optimal basis.
The PCPABF project is related to complexity and establishing limits on approximation algorithms. As a serendipitous result of research performed within the framework of this ERC Advanced Grant, the PI Muli Safra very recently established a previously unforeseen connection between Lattices and Lee Algebras. This, in turn, led Safra to a series of observations: some have to do with opportunities to improve on LLL that had not been previously explored; others, which are even more fundamental, have to do with opportunities to identify subsets of components within a set of lattice basis vectors and create algorithms that leverage that information to make improvements, both within the individual subsets and across them.
Building on recent advances that has identified promising refinements to classical lattice reduction techniques, the PCPABF team works on prototyping a new generation of lattice basis reduction algorithms. The improvements will be along two critical dimensions: Quality of reduction – producing basis vectors that are significantly shorter and more orthogonal than those obtained by existing approximation algorithms, thereby enhancing the efficiency and reliability of applications that rely on BR; and Computational efficiency – reducing run-time complexity to enable practical deployment of BR in real-time scenarios. This work is still in progress; a preprint is expected to be ready within next four months and PCPABF grant is acknowledged.
Building on recent advances that has identified promising refinements to classical lattice reduction techniques, the PCPABF team works on prototyping a new generation of lattice basis reduction algorithms. The improvements will be along two critical dimensions: Quality of reduction – producing basis vectors that are significantly shorter and more orthogonal than those obtained by existing approximation algorithms, thereby enhancing the efficiency and reliability of applications that rely on BR; and Computational efficiency – reducing run-time complexity to enable practical deployment of BR in real-time scenarios. This work is still in progress; a preprint is expected to be ready within next four months and PCPABF grant is acknowledged.