Prof. Emo Welzl and Prof. Bernd Gärtner
Theory of Combinatorial Algorithms
Teaching and Research Group Emo Welzl
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(Financed by the Innovedum Fund at ETH Zurich). We set up a new course in which students at ETH collaborate with students of other international universities on theoretical or applied projects in the area of topological data analysis. Topological data analysis is a topic which has seen a rapid increase in importance in the previous years and has found many applications.
We collaborate with 2-3 other universities on a new course in which students work in an international team on projects in the area of topological data analysis. These projects could be of theoretical nature, e.g., the design and analysis of a new algorithm, or applied, such as using methods from topological data analysis on concrete data sets. This new international course is complemented by a new local introductory course, where the necessary background is taught. In this project, students not only get to learn important skills and knowledge from topological data analysis, but also gain experience at group work with international partners, a skill that becomes more and more important in a globalized world. By collaborating with other universities, students at ETH further get to work with world- renowned experts in topological data analysis.
(Financed by the Swiss National Science Foundation). A unique sink orientation (USO) is an orientation of the n-dimensional hypercube graph, with the property that every face (subcube) has a unique sink. In particular, there is a unique global sink.
The algorithmic problem associated with a USO is that of finding the global sink. A vertex evaluation oracle can be asked for the orientations of all n edges incident to a given vertex. The complexity of a (randomized) sink-finding algorithm is the (expected) number of oracle queries it needs in the worst case to find the global sink of an n-dimensional USO. It is unknown whether the sink can be found with polynomially (in n) many oracle queries, but there are also no hardness results.
USOs have been introduced in the context of linear complementarity problems by Stickney and Watson, and later as combinatorial objects by Szabó and Welzl. Other optimization problems have also been shown to give rise to USOs ,most notably linear programming (LP), but also quadratic programming and more general convex optimization problems such as finding the smallest enclosing ball of a set of points, or a set of balls.
Being able to find the sink in polynomial time would answer two major open questions in optimization theory: (i) is there a strongly polynomial-time algorithm for linear programming? This question is on Smale's 1998 list of mathematical problems for the next century; (ii) is there a polynomial-time algorithm for P-matrix linear complementarity problems? These problems are well-studied and known to fall into a number of (recent) complexity classes, but hardness results are not known, either.
The fact that the aforementioned open questions are long-standing indicates that the goal of polynomial-time sink-finding in USOs is not very realistic. Despite this, USOs have been studied by many researchers (including the applicant) over the past twenty years, since they provide a clean and simple combinatorial framework in which many other relevant questions can be asked (and actually answered). These questions often impact or relate to concrete questions in optimization theory. For example, the currently best deterministic algorithm for P-matrix linear complementarity problems is USO-based. Notwithstanding the research interest in sink-finding in USOs, most of the initial algorithmic results of Szabó and Welzl have still not been improved significantly.
Over the last three years, many new but somewhat "isolated" insights on USOs have been obtained, although with some surprising connections showing up here and there. Much of this work was done in joint research of the applicant with students, in the context of Bachelor's, Master's and PhD theses. The goal of this project is to unify and extend this research, systematically explore promising directions that have so far only been touched, and eventually make progress also on the algorithmic problem. As Szabó and Welzl (lightheartedly, in hindsight) put it in 2001, "there is ample space for improvement". During the project, we want to explore this space.
Contact: B. Gärtner
Duration: Oct 2021 – Feb 2025.