Periodic Reporting for period 5 - ENGAGES (Next generation algorithms for grabbing and exploiting symmetry)
Berichtszeitraum: 2024-04-01 bis 2025-02-28
The ERC project EngageS (Next Generation Algorithms for Grabbing and Exploiting Symmetry) addressed the challenge of detecting and efficiently using symmetries in computational settings. Although symmetry is a powerful concept in mathematics and computer science, our theoretical understanding of the complexity of symmetry-related algorithmic problems is far from comprehensive. At the same time, existing software tools for symmetry detection are often unsuitable for synergies with modern computing hardware. EngageS investigated both the theoretical and practical aspects of algorithms involving symmetry in its various forms. This included fundamental problems such as graph isomorphism, canonization (the computation of normal forms), efficient exhaustive isomorph-free generation, and the classification of the structure of symmetries that arise in applied contexts.
Symmetries arise across a broad spectrum of real-world applications, ranging from machine learning and computer graphics to chemical databases and beyond. Effectively leveraging symmetry can significantly reduce computational effort, sometimes making seemingly intractable algorithmic problems feasible. For instance, in many search or optimization tasks, symmetric portions of the search space can be ignored once one representative has been examined. Faster symmetry detection leads to faster overall computation times. Conversely, in areas like machine learning or optimization, we might modify the object to remove symmetries (symmetry breaking) in order to improve algorithmic performance. In both cases, efficient symmetry handling is critical to enabling better algorithms.
The goal of the project was therefore to develop the next generation symmetry algorithms. Overall, its main objectives were to: 1. advance the theoretical understanding of the graph isomorphism problem by investigating its complexity, 2. design the next generation of symmetry detection algorithms suitable for modern hardware, and 3. bridge the long-standing gap between theory and practice in the area of symmetry detection and exploitation.
Substantial progress was achieved in each of these three areas over the course of the project. On the theoretical side, the project produced definitive answers to previously unresolved questions about the symmetry of combinatorial structures and the complexity of computing them. On the practical side, the project led to the development of new software libraries, now available as open-source tools. These enable more efficient symmetry detection and exploitation in real-world applications. By investigating the theoretical and practical challenges in parallel, the project brought together theoreticians and practitioners, facilitating productive collaborations that allowed advances in one domain to inform and accelerate progress in the other.
Symmetries arise across a broad spectrum of real-world applications, ranging from machine learning and computer graphics to chemical databases and beyond. Effectively leveraging symmetry can significantly reduce computational effort, sometimes making seemingly intractable algorithmic problems feasible. For instance, in many search or optimization tasks, symmetric portions of the search space can be ignored once one representative has been examined. Faster symmetry detection leads to faster overall computation times. Conversely, in areas like machine learning or optimization, we might modify the object to remove symmetries (symmetry breaking) in order to improve algorithmic performance. In both cases, efficient symmetry handling is critical to enabling better algorithms.
The goal of the project was therefore to develop the next generation symmetry algorithms. Overall, its main objectives were to: 1. advance the theoretical understanding of the graph isomorphism problem by investigating its complexity, 2. design the next generation of symmetry detection algorithms suitable for modern hardware, and 3. bridge the long-standing gap between theory and practice in the area of symmetry detection and exploitation.
Substantial progress was achieved in each of these three areas over the course of the project. On the theoretical side, the project produced definitive answers to previously unresolved questions about the symmetry of combinatorial structures and the complexity of computing them. On the practical side, the project led to the development of new software libraries, now available as open-source tools. These enable more efficient symmetry detection and exploitation in real-world applications. By investigating the theoretical and practical challenges in parallel, the project brought together theoreticians and practitioners, facilitating productive collaborations that allowed advances in one domain to inform and accelerate progress in the other.
EngageS addressed questions ranging from theoretical to practical aspects of symmetry, with an emphasis on the interplay between the two. We highlight key outcomes:
1. For the first main objective, we considered descriptive complexity theory, a field exploring the limits of efficient computation using formal logic. Here we investigated a long-standing problem regarding a logic that captures polynomial time. It asks us to find a query language in which we can express exactly all database queries that can be efficiently evaluated. Such query languages are only known for databases whose symmetries have been artificially removed. There were two candidate logics: choiceless polynomial time (CPT) and rank logic. We showed that CPT captures P-Time on certain databases. In contrast, we proved that rank logic does not capture P-Time. Further results concern counting logics related to the Weisfeiler-Leman algorithm (WL). We proved bounds on the dimension, computational complexity, and iteration number of the algorithm. We also characterized symmetry groups of graphs of bounded Hadwiger number and classified finite highly symmetric (i.e. homogeneous) graphs with colors.
2. For the second main objective, we researched and developed modern methods for detecting symmetries and computing isomorphisms, leading to the creation of a new randomized algorithm and the corresponding software tool “dejavu”. This algorithm can be used to compute all symmetries of arbitrary, explicitly given combinatorial objects, including graphs, networks, and groups. To complement this, we also developed a preprocessor “sassy” that improves the efficiency of our solver and also improves the efficiency of existing solvers.
The project also worked towards breaking the black-box approach to symmetry detection. Typically, symmetry detection and resolution of computational tasks using mathematical solvers are considered separately. In the context of satisfiability solving (SAT), we devised an approach that integrates symmetry detection directly into the solving process. This approach outperforms approaches that treat the two tasks separately. We also developed a new tool for symmetry breaking, which simplifies problem instances through symmetry use.
3. The third main objective was achieved by a theoretical analysis of the existing and newly developed practical solvers. Our theoretical results show that various aspects of the algorithm are provably near-optimal. On the other hand, the design of the symmetry breaking software was guided by theoretical analysis of the underlying algorithmic problems. We also bridged the gap by a holistic approach to symmetry detection and exploitation in SAT solving and mixed integer programming. This resulted in software tools with theoretical guarantees and simultaneously with an algorithm engineering focus on practical instances.
All developed software developed has been released as open-source. Many theoretical results are spread out over the more than 50 peer-reviewed publications. (See project website.)
1. For the first main objective, we considered descriptive complexity theory, a field exploring the limits of efficient computation using formal logic. Here we investigated a long-standing problem regarding a logic that captures polynomial time. It asks us to find a query language in which we can express exactly all database queries that can be efficiently evaluated. Such query languages are only known for databases whose symmetries have been artificially removed. There were two candidate logics: choiceless polynomial time (CPT) and rank logic. We showed that CPT captures P-Time on certain databases. In contrast, we proved that rank logic does not capture P-Time. Further results concern counting logics related to the Weisfeiler-Leman algorithm (WL). We proved bounds on the dimension, computational complexity, and iteration number of the algorithm. We also characterized symmetry groups of graphs of bounded Hadwiger number and classified finite highly symmetric (i.e. homogeneous) graphs with colors.
2. For the second main objective, we researched and developed modern methods for detecting symmetries and computing isomorphisms, leading to the creation of a new randomized algorithm and the corresponding software tool “dejavu”. This algorithm can be used to compute all symmetries of arbitrary, explicitly given combinatorial objects, including graphs, networks, and groups. To complement this, we also developed a preprocessor “sassy” that improves the efficiency of our solver and also improves the efficiency of existing solvers.
The project also worked towards breaking the black-box approach to symmetry detection. Typically, symmetry detection and resolution of computational tasks using mathematical solvers are considered separately. In the context of satisfiability solving (SAT), we devised an approach that integrates symmetry detection directly into the solving process. This approach outperforms approaches that treat the two tasks separately. We also developed a new tool for symmetry breaking, which simplifies problem instances through symmetry use.
3. The third main objective was achieved by a theoretical analysis of the existing and newly developed practical solvers. Our theoretical results show that various aspects of the algorithm are provably near-optimal. On the other hand, the design of the symmetry breaking software was guided by theoretical analysis of the underlying algorithmic problems. We also bridged the gap by a holistic approach to symmetry detection and exploitation in SAT solving and mixed integer programming. This resulted in software tools with theoretical guarantees and simultaneously with an algorithm engineering focus on practical instances.
All developed software developed has been released as open-source. Many theoretical results are spread out over the more than 50 peer-reviewed publications. (See project website.)
1. We proved rank logic does not capture P-Time. This resolves a long-standing open question in descriptive complexity theory. We also obtained the currently best bounds on the number of variables required in fixed-point logic with counting, and the first superlinear bounds on quantifier depth, equivalently on the iteration number of WL. Our lower bound construction has already been exploited in proof complexity. We also resolved three conjectures posed by László Babai concerning the relationship between the symmetry structure of graphs and the substructures they contain (specifically in terms of graph minors). The proofs combined a surprisingly diverse set of techniques involving Archimedean geometries, sphere packings, and graph covers. We also classified homogeneous graphs with colors (unary predicates).
2. Towards the next generation of algorithms, the new symmetry algorithm and its corresponding software library “dejavu” developed in this project are now the state-of-the-art and currently fastest tools available for symmetry detection. Although randomized, the algorithm can effectively compute the symmetry groups of arbitrary combinatorial objects. Similarly, the symmetry breaking tool “satsuma” is currently state-of-the-art for the removal of symmetries in SAT-formulas.
3. With regard to bridging the gap between theory and practice, these results and advances were made possible by bringing together a team with diverse but complementary expertise, combining theoretical insight, algorithmic design, and practical implementation.
2. Towards the next generation of algorithms, the new symmetry algorithm and its corresponding software library “dejavu” developed in this project are now the state-of-the-art and currently fastest tools available for symmetry detection. Although randomized, the algorithm can effectively compute the symmetry groups of arbitrary combinatorial objects. Similarly, the symmetry breaking tool “satsuma” is currently state-of-the-art for the removal of symmetries in SAT-formulas.
3. With regard to bridging the gap between theory and practice, these results and advances were made possible by bringing together a team with diverse but complementary expertise, combining theoretical insight, algorithmic design, and practical implementation.