DIMACS REU 2026
I am a graduate mathematics student at Bar-Ilan University participating in the DIMACS REU summer research program at Rutgers University. My academic interests center on theoretical computer science.
During this program, I am collaborating on analyzing performance improvements in classical online routing problems when probabilistic samples on the input is provided.
Research Scope
Given an instance of the Steiner Forest problem, we study whether access to a probabilistic sample can allow for better algorithmic performance. We assume the model is given a random \(p\)-fraction of the problem terminals up front as a sample, on which the algorithm can calculate arbitrary preprocessing constraints.
The remaining terminals are then revealed sequentially in an online fashion. The objective is to design an approximation algorithm that scales gracefully, achieving a competitive ratio of \(\log (1/p)\) to smooth the gap between fully offline and traditional online bounds.
Prof. Roie Levin · Prof. Arnold Filtser · Ryan Liu
Weekly Progress
Reviewed the problem statement as well as classical methods of solving online and offline Steiner Forest and Steiner Tree problems. Discussed the research scope and potential directions with mentors. Main tools for the problem include linear programming (LP) relaxations, primal-dual frameworks, and greedy algorithms.
Presented our preliminary project setup to the REU cohort. Held comprehensive group synchronization meetings to outline current active thresholds across online Steiner Forest variations. Evaluated structural bounds where basic greedy approaches drop edge components, mapping edge-case behaviors to pin down exact directions for proof extensions. Continued reading on more advanced materials regarding offline Steiner Forest approximations and their potential applications to our problem.
Analyzed candidate analysis of using the sample to break up the problem into smaller, more manageable subproblems. This direction ended up as a failure, which highlighted the specific difficulty in the Steiner Forest problem being not only a networking problem, but a clustering problem as well.
Proved a simple lemma showing that if clustering was known in advance, the problem would be equivilent to Steiner Tree. Further maintaing that its novel difficulty lies in the interplay between clustering and network optimization. Reviewed recent graph formulations regarding offline 1.994-approximations to adapt localized network constraints to our model.
Proved a 1/p-competitive algorithm for the online Steiner Tree problem with a sample, in a novel approach that limits the number of dual ball intersections. This approach may be extended to the Steiner Forest problem, but presents new challenges for the clustering aspect of the problem. Further research is needed to determine if this approach can be adapted to the Steiner Forest problem.
Ryan Liu came up with a new approach for analysis of Steiner Forest, utilizing a cost-sharing scheme to attribute cost to each individual terminal pair. This indeed yields a 1/p-competitive algorithm for the online Steiner Forest problem with a sample, and is a promising direction for further research. The next steps are to consider improvements for a log(1/p)-competitive algorithm, and perhaps generalize this tool for different set cover problems.
Credits
Thank you to Prof. Roie Levin, Dr. Lazaros Gallos, and Lawrence Frolov for their ongoing guidance and logistical coordination throughout the program.
This research is conducted under the auspices of the DIMACS REU program at Rutgers University.