Email: cerny@kam.mff.cuni.cz
Home Institution: Department of Applied Mathematics, Charles University
Institution: Rutgers University – Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
Dates: May 27 – July 18, 2025
Supervisor: Dr. Arpita Biswas
Collaborator: Hana Salavcová
Project Title: Fair Allocation Problem
Personal Website: Here
Curriculum Vitae: CV
Our project focuses on the fair allocation of indivisible goods when the standard assumption of exclusivity is relaxed. Specifically, we study a model where each item can be shared among up to k agents—a framework known as k-sharing. This setting captures a more flexible and realistic approach to fair division, where goods like digital assets, shared time slots, or communal resources may not be limited to a single owner.
This week, I focused on building foundational knowledge in fair allocation. I read the following papers:
These readings introduced me to various fairness notions like EF1, EFX, and MMS. The Amanatidis et al. paper provided a broad overview of the current landscape in indivisible fair division, including algorithmic challenges and impossibility results. The survey by Aziz et al. had a significant overlap with the first, but also included results about chores and gave detailed descriptions of the most standard algorithms in the area. The chapter by Biswas et al. summarized the theoretical foundations relevant to the constrained settings we are likely to explore in our project.
This week, we finalized the definition of our research problem: fair allocation of indivisible goods with fair sharing constraints. We clarified our working assumptions and terminology through discussions with our supervisor, and we began to examine the relationship between our model and existing frameworks in the literature.
In addition to reviewing earlier papers, I read the following new works to better understand the theoretical underpinnings of MMS allocations and impossibility results:
This week, we worked on defining fairness concepts for shared allocations, studied tight negative examples for MMS guarantees, and began analyzing structural conditions under which MMS allocations are guaranteed.
Reading the recent WINE paper by Feige et al. gave us valuable insights into the limitations of MMS guarantees and prompted us to investigate how such constructions translate to shared ownership. Our proposed upper bound for guaranteed MMS allocations shows promise but raises deeper questions, especially as the number of agents grows. Discussions with colleagues outside the REU group brought useful perspective and may inform new approaches to bounding techniques.
We investigated a surprising connection between our fair allocation problem and results from coding theory, particularly covering codes. We also began planning a potential research paper submission.
The link to coding theory was unexpected but highly promising—it gave us a fresh lens through which to analyze constraints on fair allocations. The odd/even discrepancy in MMS existence is particularly intriguing and may point to new impossibility results.
This week brought a significant development: while completing the draft of our paper, our supervisor informed us of a newly published paper whose main result directly subsumes the core contribution of our own manuscript. After a close reading, we confirmed that the result indeed renders the main part of our work redundant. As a result, we decided not to proceed with our submission and instead shifted our focus toward new research directions.
The discovery that a newly released paper proved the main result of our own work was strongly disappointing and frustrating. After investing considerable time into developing what we believed was a novel contribution, it was difficult to accept that the core of our draft had already appeared in the literature just months earlier.
At the same time, reading the paper was also encouraging: many of its key ideas closely mirrored our own, suggesting that we had independently arrived at strong and relevant results in the field. While it rendered our submission obsolete, it validated the originality and quality of our thinking. This gave us confidence that we are capable of contributing meaningful answers to current open questions in fair division.
We revisited the recent paper by Barman, Rammohan, and Sethia in detail to understand how their framework relates to our setting. Building on this, we made systematic comparisons of the SSMS values across different sharing models with varying cost structures. Finally, we proposed a probabilistic method to estimate the minimum number of goods K needed to guarantee the existence of an MMS allocation in models with bounded maximum item cost.
The comparisons across models helped establish a week relation between the fairness guarantees when cost assumptions vary. The probabilistic method builds on a previously considered tool for understanding when MMS can be expected to hold, even in complex settings, however, inspired by the manuscript by Bearman et al., our analysis was better.
We developed a modification of the Bag-Filling algorithm to approximate MMS allocations in the equal-share model. Using the structural relationships between MMS and SMMS values across cost-sharing models—established last week—we showed that the same algorithm also yields exact MMS allocations in more general models with bounded maximum cost.
This week marked a clear transition from theory to writing. Having both theoretical guarantees and an algorithm in hand, we now have a more-or-less complete result ready for publication.
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I would like to gratefully acknowledge the guidance of my supervisor Arpita Biswas throughout the course of this project.
I would also like to thank the DIMACS REU 2025 program for providing me with this incredible research opportunity. I am especially grateful to Rutgers University and the DIMACS center for hosting the program and creating a stimulating and supportive research environment.
I am further partly supported by: