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   https://doi.org/10.1215/00127094-2022-0041  

   Balmer, Paul ; Gallauer, Martin ( February 2023 , Duke Mathematical Journal)
2. Local Frechet Permutation

   Perry, Jonathan ; Raichel, Benjamin ( July 2024 , Proceedings 36th Canadian Conference on Computational Geometry)
3. Resolutions by permutation modules

   https://doi.org/10.1007/s00013-020-01493-w  

   Balmer, Paul ; Benson, Dave ( November 2020 , Archiv der Mathematik)

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4. Permutation modules and cohomological singularity

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   Balmer, Paul ; Gallauer, Martin ( January 2022 , Commentarii Mathematici Helvetici)
5. Bayesian Optimization over Permutation Spaces

   https://doi.org/10.1609/aaai.v36i6.20604  

   Deshwal, Aryan ; Belakaria, Syrine ; Doppa, Janardhan Rao ; Kim, Dae Hyun ( June 2022 , Proceedings of the AAAI Conference on Artificial Intelligence)

   Optimizing expensive to evaluate black-box functions over an input space consisting of all permutations of d objects is an important problem with many real-world applications. For example, placement of functional blocks in hardware design to optimize performance via simulations. The overall goal is to minimize the number of function evaluations to find high-performing permutations. The key challenge in solving this problem using the Bayesian optimization (BO) framework is to trade-off the complexity of statistical model and tractability of acquisition function optimization. In this paper, we propose and evaluate two algorithms for BO over Permutation Spaces (BOPS). First, BOPS-T employs Gaussian process (GP) surrogate model with Kendall kernels and a Tractable acquisition function optimization approach to select the sequence of permutations for evaluation. Second, BOPS-H employs GP surrogate model with Mallow kernels and a Heuristic search approach to optimize the acquisition function. We theoretically analyze the performance of BOPS-T to show that their regret grows sub-linearly. Our experiments on multiple synthetic and real-world benchmarks show that both BOPS-T and BOPS-H perform better than the state-of-the-art BO algorithm for combinatorial spaces. To drive future research on this important problem, we make new resources and real-world benchmarks available to the community. 

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