Prof. Dr. Tobias Friedrich

# Testing for Geometry in Random Satisfiability Instances

## Motivation

SAT is the archetypal NP-complete problem and is used as a starting point for hardness reductions in theory, but in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. One possible explanation for this disparity is that industrial SAT instances have particular properties that make them more easily solvable. One approach to understanding this difference between theory and practice is to study random SAT models whose generated instances share structural properties with real-world SAT instances. This approach has been successfully applied in network problems, where geometric random graph models have been shown to share many properties with real-world networks, be it structural or algorithmic. Two industrial SAT instances (left and middle) and a geometric SAT instance (right). In a recent paper [2] we studied how two properties of random SAT instances affect their tractability. More importantly we showed that when SAT instances are generated with a particular underlying geometry, they can be easily refuted, while this is not the case when the geometry is removed. While in the non-geometric mode each variable is added to each clause independently according to its weight, the geometric model affects the correlation between its clauses as follows. Variables and clauses are dropped uniformly at random as points in a compact space, and then each clause connects to its k closest variables. This way clauses that are geometrically close are likely to contain the same variables. While this is significant progress in explaining the tractability of SAT instances, it does not provide us with a complete answer. The reason is that this geometric random SAT model generates instances that are "too easy" compared to the real-world ones. That is, generated SAT formulae have constant-size unsatisfiable subformulae, while real-world instances have logarithmic-size ones.

## The Goal of this Project

The goal of the project is to study the structural properties of geometric SAT instances and compare them to those of real-world instances. This could be done in several ways:

1. Obtaining bounds on the total variation distance between the geometric random k-SAT distribution and some basic/non-geometric generation procedure, such as uniform random connection.

2. Developing (with proof) a statistical test for whether a given instance has underlying geometry. Possible test statistics could include degree distribution, clustering coefficient, diameter, and so on.

3. Experimentally finding a test statistic for differentiating between real-world SAT instances and geometric random k-SAT instances; for recent examples, see [1, 3].

It would also be interesting to understand how any of these analyses depend on the dimension of the geometric space.

### What we expect from you.

We expect students who are familiar with graph theory and algorithms, enjoy writing proofs, and are interested in learning more about probability theory. We understand that you won't know all the background, and we'll provide you with sources for learning. You should bring the willingness and curiosity to delve into an interesting interdisciplinary research topic in theoretical computer science, which makes use of tools from several domains of mathematics.

### What you can expect from us.

We will gently introduce you to the field and accompany you all along this interesting journey. This will be a team effort, and we aim at publishing our results at a renowned international conference.

 [1] P. Almagro, M. Boguña, M. Ángeles Serrano (2022): Detecting the ultra low dimensionality of real networks. [2] T. Bläsius, T. Friedrich, A. Göbel, J. Levy, R. Rothenberger (2020): The impact of heterogeneity and geometry on the proof complexity of random satisfiability. [3] T. Friedrich, A. Göbel, M. Katzmann, L. Schiller (2023): A simple statistic for determining the dimensionality of complex networks.

# Project Team

The Master Project is organized by the Algorithm Engineering Group. The following group members are participating:

## Prof. Dr. Tobias Friedrich

Supervisor

Chair for Algorithm Engineering
Hasso Plattner Institute

Office: K-2.15
E-Mail: Friedrich(at)hpi.de

# Andreas Göbel

Chair for Algorithm Engineering
Hasso Plattner Institute

Office: K-2.06
Tel.: +49 331 5509-424
E-Mail: Andreas.Goebel(at)hpi.de

## Samuel Baguley

Chair for Algorithm Engineering
Hasso Plattner Institute

Office: K-2.06
Tel.:
E-Mail: Samuel.Baguley(at)hpi.de

## Lukas Weyand

Bachelor Project 2020/21
Bachelor Thesis 2021
Preparation of Real Estate Data for Deep Learning

Chair for Algorithm Engineering
Hasso Plattner Institute

## Jessica Dierking

Scientific Programmer
Tutor Algorithmic Problem Solving WS 2020/21
Tutor Mathematik II SS 2021
Bachelor Project (TomTom) 2021/22
Tutor Theoretische Informatik II SS 2022
Bachelor Thesis 2022
Modelling Traffic Flow for Electric Vehicles

Chair for Algorithm Engineering
Hasso Plattner Institute

## Clemens Schielicke

Chair for Algorithm Engineering
Hasso Plattner Institute

E-Mail:

## Tobias Sträubig

Chair for Algorithm Engineering
Hasso Plattner Institute

E-Mail: