We briefly summarize the progress, grouped by topic. Several advances made throughout this project have not been published yet. Below, we focus on published results.
Online Optimization and Streaming Algorithms
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Advances have been made on several fronts in this area. We made progress on an online rounding procedure with quite broad applications and were able to show that prior approaches popular within Mechanism Design are not strong enough to provide comparable results. Our online rounding procedure has interesting implications to problems related to the Matroid Secretary Problem. Also, we presented the first constant-competitive variant of a non-trivial and multiply studied variant Matroid Secretary Problem, without knowing the matroid upfront. Moreover, we have been able to shed light also on the maximization of so-called (monotone) submodular functions in the streaming setting. In particular, we have been able to settle the approximability of canonical problems in this area through algorithmic and hardness results based on novel ideas.
Problems related to Integer Programming with Bounded Subdeterminants
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Major hurdles have been identified to tackle the central open question in the area, namely whether integer programs (IPs) with bounded subdeterminants can be solved efficiently. We particularly focussed on so-called congruency-constrained combinatorial optimization problems. Here, we significantly expanded prior knowledge by exploiting connections to other mathematical fields, including Additive Combinatorics and results from Number Theory. This resolves several questions raised in the proposal. One key goal was to advance beyond what people call the bimodular case in the congruency-constrained setting, and we have been able to make multiple progress on this front. This showed for the first time techniques of a broader applicability beyond the bimodular case in relevant settings. Further very recent progress, identifies important limits of approaches suggested by other leading researchers, which helps in directing future research on this notoriously difficult topic.
Further problem classes
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Moreover, we made progress on several other well-known problems, where novel Combinatorial Optimization techniques, in the spirit of this proposal, helped to make significant progress on long-standing open problems. In particular, we have been able to close the gap in terms of approximability between the famous Traveling Salesman Problem and its path version, which are two heavily studied problem in the field. We also made significant progress on several Network Design problems. This includes called augmentation problems, where we obtained the first better-than-2 approximation for Weighted Tree Augmentation; a problem that has been open for decades despite extensive research. Our result improves on a large number of prior papers by leading researcher. Also, we settled the approximability of certain k-edge-connected subgraph probelms, which are very classic network design problems, and made further important progress on the so-called unsplittable flow problem. Moreover, we developed a novel viewpoint to resolve an open question in colorful clustering.
