Ryan Liu

My week started with move in and meeting my mentor for the first time. I attended orientation and met everyone, and we listened to two short seminar talks. With my mentor, we discussed some of the algorithms and analyses used in the various papers I was given to read. I presented the \(\log k\)-competitive ratio of the augmented greedy algorithm for online Steiner Forest to Roie and Gal. Through some readings and practicing, I also got more comfortable with LPs and primal-dual analysis in general.

• Notes on the augmented greedy algorithm for Online Steiner Forest
• Notes on the greedy algorithm for Online Steiner Tree
• Steiner Forest 2-approximation
• Online Steiner Tree With a Sample

We made and presented our project to everyone else on Tuesday, and also got to hear about the cool stuff everyone else is going to do. All of the collaborators met for the first time together in person, and we discussed some of the current results in online Steiner Forest and Steiner Tree. Spent some time coming up with ideas and noting down where they failed as potential directions in taking our proof.

I went with a few other students to the Computational Geometry Week 2026 hosted by DIMACS. There was an interesting talk by Prof. Ronitt Rubinfeld (MIT) on sublinear algorithms and the new directions currently being taken.

I came up with a potential algorithm and proved that it has \(\log 1/p\)-competitive ratio on a nice subset of solution terminals. However it has a lot of issues which I'm not sure can be overcome with modifications to the algorithm that prevents it from achieving the (assumed) optimal solution. A potential direction is to read up on the set cover problem and see if there are any interesting ideas that can be adapted to our problem.

I proved a small lemma showing that the augmented greedy algorithm achieves a \(\log 1/p\)-competitive ratio on connected components in the optimal solution that have no samples. The initial ideas we had for the set cover problem seemed to not be feasible, so we read up on the the new 1.994-approxiamtion for the offiline Steiner Forest. It contained a lot of cool ideas such as Autarkic pairs and \(\epsilon\)-moat growing that could be useful for the offline step of our algorithm.

• 1.994-approximation for offline Steiner Forest

While tinkering with some different ideas, I found an interesting concept of cost-sharing used primarily in economics but seemed to have good applications to our problem. Using this idea, I was able to prove a \(1/p\)-competitive ratio for essentially any problem using the online with-a-sample framework independently of the structural properties of the problem. The next step is likely leveraging specific ideas about Steiner Forest to obtain the log ratio.

• Cost sharing paper

I presented my proof to the group and we scrutinized to see if it passed the smell test. I further refined the analysis of for the \(1/p\)-ratio proof, since it uses some pretty neat tricks with expectation and probability that needs to be accurately defined. I also started reading more about \(\beta\)-strict cost sharing and how they manage to obtain constant strictness. Some of the analysis ideas could be useful in improving our algorithm or analysis.

• Simplified and improved cost-sharing scheme

Did some more reading and iterating to see if we could improve our solution. The main issue is that most of the preexisting work lives within the offline framework, which is useful when analyzing an algorithm. However, taking those ideas and implementing them into an online solution makes it harder, since a lot of the structures used in the proof are highly unstable to slight changes in the problem. I spent a while thinking about different ways to modify notions of "progress" or bounding cost in a way that was more stable.