Martin Belluš

Our group successfully arrived at Rutgers, after an 8-hour flight and a 3-hour journey from the airport. I had my first meeting with my mentor. I read through several papers to understand the foundations of this project and prepare for my initial presentation.

I also attended the student orientation, but didn't learn many names. That changed the next day, when a group of us met outside the dorms and threw a frisbee in a circle. Allison also invited me to a pickup frisbee practice, which was fun but also really exhausting.

I began the week by meeting with my mentor to discuss the details of my project and my initial presentation . I finally finished the presentation and gave a talk about my project to my fellow REU participants. This week I focused on understanding the results and techniques used in the following papers:

• Neoh et al. (2026). Online Fair Division with Additional Information https://arxiv.org/abs/2505.24503
• Zhou et al. (2023). Multi-agent Online Scheduling: MMS Allocations for Indivisible Items https://arxiv.org/abs/2304.13405

I also set up an Overleaf document to share progress with my mentor and made preliminary observations about the model.

My week started by a long meeting with my mentor, where we finalized description of the model, which I will be working with and outlined possible directions I can focus on in the project. I started adapting an algorithm from the paper by Zhou et al. (2023) to our model with a small number of agents. I look forward to discussing the results with my mentor at our next meeting to see whether the approach can be extended to more agents.

In terms of social activities, this week was a bit quieter. However, I once again attended the pickup frisbee practice on Sunday and watched the F1 race afterwards.

After the Tuesday meeting with my mentor, I found a flaw in the proof of the competitiveness of the algorithm for a small number of agents. Thus the majority of the week was spent rewriting the proof. It turned out the theoretical guarantees of the algorithm were weaker than I originally thought, but the result is still somewhat interesting. I also started thinking about how the algorithm can be extended to more agents.

Our Prague group (Sofia, Jakub, and me) finally went to New York City on Friday. We walked around the Manhattan borough and went to the Museum of Modern Art, where we spent 3 hours. I tried a food truck for the first time (would definitely recommend) and bought an incredible amount of bagels for just $6. I also played poker for the first time, hosted by Eli.