Periodic Reporting for period 4 - ICOPT (Fundamental Problems at the Interface of Combinatorial Optimization with Integer Programming and Online Optimization)
Período documentado: 2024-05-01 hasta 2024-10-31
Algorithms are omnipresent in our everyday lives, and the need to understand how to solve large problems fast is ever-increasing, in particular with the advent of Big Data. Whenever a real-world problem of computational nature emerges, before it can be tackled by algorithms, it is transformed into a well-defined computational problem. Typically, there are many ways to transform (or model) an often fuzzily defined real-world problem into a formal computational one, and it is crucial to find models that both accurately reflect the reality and can be solved quickly. This leads to several natural questions:
- What can be computed efficiently?
- What are the underlying reasons why some computational problems can be solved fast?
- How powerful are different classes of mathematical optimization models?
The surge of computational problems during the last decades has led to a multitude of communities working on the design of fast algorithms specialized to different settings. Over the years, each field has developed their own standard toolbox for approaching algorithmic questions. However, not surprisingly, breakthrough results were often achieved by combining approaches from different fields.
The goal of this proposal is to leverage and significantly extend techniques from the field of Combinatorial Optimization to address some fundamental, open algorithmic questions in other, related areas, which are within the realm of the three above-highlighted questions. To this end, we identified different research thrusts, which, despite the fact that they cover problems from traditionally different algorithmic communities, share many combinatorial features. The areas we consider include problems in Integer and Online Optimization, and our focus is on long-standing open problems, where new connections to Combinatorial Optimization justify the hopes that substantial progress within the realm if this project is possible. These hopes are moreover fueled by the fact that the cross-disciplinary approaches we suggest are beyond the well-trodden paths.
Our goal is to obtain advances on foundational computational questions with the goal to reveal new algorithmic techniques, whose efficacy is corroborated by a rigorous theoretical analysis. This type of foundational results aim at being a first crucial step toward the possibility of obtaining fast ways to solve relevant computational problems that cannot be solved with current methods within any reasonable computational time.
- What can be computed efficiently?
- What are the underlying reasons why some computational problems can be solved fast?
- How powerful are different classes of mathematical optimization models?
The surge of computational problems during the last decades has led to a multitude of communities working on the design of fast algorithms specialized to different settings. Over the years, each field has developed their own standard toolbox for approaching algorithmic questions. However, not surprisingly, breakthrough results were often achieved by combining approaches from different fields.
The goal of this proposal is to leverage and significantly extend techniques from the field of Combinatorial Optimization to address some fundamental, open algorithmic questions in other, related areas, which are within the realm of the three above-highlighted questions. To this end, we identified different research thrusts, which, despite the fact that they cover problems from traditionally different algorithmic communities, share many combinatorial features. The areas we consider include problems in Integer and Online Optimization, and our focus is on long-standing open problems, where new connections to Combinatorial Optimization justify the hopes that substantial progress within the realm if this project is possible. These hopes are moreover fueled by the fact that the cross-disciplinary approaches we suggest are beyond the well-trodden paths.
Our goal is to obtain advances on foundational computational questions with the goal to reveal new algorithmic techniques, whose efficacy is corroborated by a rigorous theoretical analysis. This type of foundational results aim at being a first crucial step toward the possibility of obtaining fast ways to solve relevant computational problems that cannot be solved with current methods within any reasonable computational time.
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.
Online Optimization and Streaming Algorithms
----------------------------------------------------------------------
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
----------------------------------------------------------------------
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
----------------------------------------------------------------------
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.
On the algorithmic side, there has been progress beyond the state of the art on multiple fronts, as outlined above. We have been able to achieve several of the objectives outlined in the proposal. In particular, we made significant progress on both important variants of the matroid secretary problem and also conguency-constrained optimization. They are both key topics of our research thrust at the intersection of Combinatorial Optimization and Online Optimization/Integer Programming. The work on this ERC proposal inspired a large number of additional question on which we plan to continue in the coming years, partially with previous members of this action and also with other international collaborators.