General Information

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Project Description

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Weekly Log

Week 1:

I arrived at DIMACS and met everyone. Had my first meeting with Prof. Feigenbaum to discuss the formalism of stable matchings where the vertices are partitioned arbitrarily. Unfortunately, I got sick soon after arriving :(.

Week 2:

I found and wrote up a proof of stability for the |H|=2 case, and after some discussion figured out a graph with no stable matching for |H|=3. We discussed possible modifications of the model and many follow-up research questions.

Week 3:

We discussed whether finding if a fixed matching has no blocking coalitions can be done efficiently, and more broadly deciding if a graph admits a stable matching. We focused also on the model where randomness is incorporated. I proved the unconditional existence of stable matchings for the randomized model in the singleton partitions case (for all possible definitions).

Week 4:

I discussed a proof that finding if a fixed matching has no blocking coalition is NP-hard. I proved there exist graphs with blocking coalitions in the randomized case for arbitrary vertex partitions.
As a follow-up from last week, I also showed that a randomized blocking coalition implies a deterministic blocking coalition for singleton partition case.

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Additional Information

• My Mentor Prof. Itai Feigenbaum
• My Personal Website
• This website template is due to Andrew Xie from DIMACS REU 2025.