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.
