Periodic Reporting for period 3 - ENGAGES (Next generation algorithms for grabbing and exploiting symmetry)
Reporting period: 2021-04-01 to 2022-09-30
The ERC project EngageS (next generation algorithms for grabbing and exploiting symmetry) investigates algorithms for the detection and the efficient use of symmetries. On the theoretical side, our understanding of the complexity of algorithmic problems related to symmetry is far from comprehensive. On the practical side classical software libraries turn out to be unsuitable for synergy with modern developments in computing hardware. The project investigates both the theoretical and practical aspects of algorithms involving symmetry. This includes related aspects such as the famous graph isomorphism problem, the computation of normal forms (canonization), efficient exhaustive isomorph-free generation, and the classification of the structure of symmetries that arise in practice.
In fact, symmetries appear in many different contexts, ranging from machine learning over computer graphics to chemical data bases and beyond. A good understanding of symmetries can typically help us to speed up the solution of algorithmic problems or make them feasible in the first place. Intuitively when an algorithm is searching for a solution, if two parts of the search space are symmetric only one of the parts has to be investigated. Faster symmetry detection leads to faster overall computation times. In other contexts we prefer to perform a modification of the object to remove the symmetries (also called symmetry breaking). This can in particular be true in contexts such as machine learning or optimization and often leads to a better general understanding of a given algorithmic task.
The goal of the project is therefore to develop next generation symmetry algorithms. Overall, the main objectives are to tackle the complexity of the graph isomorphism problem, to design the next generation of symmetry detection algorithms and to close the vast gap between theory and practice in the area of symmetry detection and exploitation.
In fact, symmetries appear in many different contexts, ranging from machine learning over computer graphics to chemical data bases and beyond. A good understanding of symmetries can typically help us to speed up the solution of algorithmic problems or make them feasible in the first place. Intuitively when an algorithm is searching for a solution, if two parts of the search space are symmetric only one of the parts has to be investigated. Faster symmetry detection leads to faster overall computation times. In other contexts we prefer to perform a modification of the object to remove the symmetries (also called symmetry breaking). This can in particular be true in contexts such as machine learning or optimization and often leads to a better general understanding of a given algorithmic task.
The goal of the project is therefore to develop next generation symmetry algorithms. Overall, the main objectives are to tackle the complexity of the graph isomorphism problem, to design the next generation of symmetry detection algorithms and to close the vast gap between theory and practice in the area of symmetry detection and exploitation.
In the first half, the project focused on various aspects involving symmetries and algorithms.
On of these aspects was the design of a software library for the detection of symmetries. This led to a new randomized algorithm and a corresponding software tool "dejavu", which is freely obtainable via the project's website. This algorithm can be used to compute all symmetries of arbitrary, explicitly given combinatorial objects.
A second focus was the question for a logic capturing P-Time. This is a famous open problem from the realm of finite model theory and in particular descriptive complexity theory. Roughly speaking the search here is for a query language that allows a user to express all database queries that can be efficiently answered by some algorithm, but no more than that. Such query languages are known to exist should the database be given in a form that actively avoids symmetries. Therefore, the treatment or removal of symmetries lies at the bottom of this tenacious open problem. Before the project there were two candidate logics believed to possibly capture P-Time, namely choiceless polynomial time (CPT) and Rank logic. Within the project it was shown that the former (CPT) captures PTime on a certain databases. In contrast to that, rank logic was ruled out as a candidate. Indeed it was rigorously shown that rank logic does not capture P-Time.
A third focus of the project was the investigation of structural symmetries of combinatorial objects. This includes decomposition techniques to reduce the task of finding symmetry in large objects to finding symmetries in smaller pieces. It also include a technical, but very precise, structural description of the symmetry groups that combinatorial objects may have depending on how they are composed. Along these lines, the concept of 2.5-connected components was developed and investigated and the structure of the symmetries of an important graph class (the graphs of bounded Hadwiger number or equivalently graphs with a forbidden clique minor) was characterized.
On of these aspects was the design of a software library for the detection of symmetries. This led to a new randomized algorithm and a corresponding software tool "dejavu", which is freely obtainable via the project's website. This algorithm can be used to compute all symmetries of arbitrary, explicitly given combinatorial objects.
A second focus was the question for a logic capturing P-Time. This is a famous open problem from the realm of finite model theory and in particular descriptive complexity theory. Roughly speaking the search here is for a query language that allows a user to express all database queries that can be efficiently answered by some algorithm, but no more than that. Such query languages are known to exist should the database be given in a form that actively avoids symmetries. Therefore, the treatment or removal of symmetries lies at the bottom of this tenacious open problem. Before the project there were two candidate logics believed to possibly capture P-Time, namely choiceless polynomial time (CPT) and Rank logic. Within the project it was shown that the former (CPT) captures PTime on a certain databases. In contrast to that, rank logic was ruled out as a candidate. Indeed it was rigorously shown that rank logic does not capture P-Time.
A third focus of the project was the investigation of structural symmetries of combinatorial objects. This includes decomposition techniques to reduce the task of finding symmetry in large objects to finding symmetries in smaller pieces. It also include a technical, but very precise, structural description of the symmetry groups that combinatorial objects may have depending on how they are composed. Along these lines, the concept of 2.5-connected components was developed and investigated and the structure of the symmetries of an important graph class (the graphs of bounded Hadwiger number or equivalently graphs with a forbidden clique minor) was characterized.
Three of the results obtained within the project go significantly beyond the state of the art. These include the development, in relation to "dejavu", of the first efficient parallel symmetry detection tool that, albeit randomized, can effectively compute symmetry groups of arbitrary combinatorial objects. The results also include a formal proof ruling out one of the two most promising approaches regarding the hunt for a logic capturing P-Time (showing that rank-logic does not do so). Finally the results include the resolution of three conjectures made by László Babai regarding the symmetry structure of graphs in terms of substructures they contain (specifically in terms of graph minors). For the latter result it was somewhat surprising that despite the techniques being diverse (involving Archimedian geometries, sphere packings, and graph covers) everything added up to a clean, comprehensive result in the end. Overall, the results were made possible by bringing together a team with diverse experiences, skills, and networks of expertise.
In the second half of the project we will focus on further bridging the gap between theory and practice, rigorously formalizing that symmetry lies at the core of the question for a P-Time logic, as well as attacking bottlenecks to improving the theoretical complexity bounds known for symmetry detection.
In the second half of the project we will focus on further bridging the gap between theory and practice, rigorously formalizing that symmetry lies at the core of the question for a P-Time logic, as well as attacking bottlenecks to improving the theoretical complexity bounds known for symmetry detection.