Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Theory of Combinatorial Algorithms
Prof. Emo Welzl and Prof. Bernd Gärtner
Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Prof. Emo Welzl and Prof. Bernd Gärtner
Theory of Combinatorial Algorithms
Teaching and Research Group Emo Welzl
Institut für Theoretische Informatik
Departement Informatik
ETH Zürich
CH–8092 Zürich
| phone | +41–44–632 73 92 |
| fax | +41–44–632 10 63 |
(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.
(Financed by the Swiss National Science Foundation). Combinatorial questions about possibly high-dimensional point sets are at the core of the research field of discrete and computational geometry, a field that lies in the intersection of mathematics and computer science and incorporates many ideas and concepts from both areas. A fundamental theorem about the combinatorics of point sets is Tverberg’s theorem, named after Helge Tverberg who presented the first proof 60 years ago. Tverberg’s theorem states that any set of (r − 1)(d + 1) + 1 points in Rd can be partitioned into r parts such that the convex hulls of the parts have a point in common.
Tverberg’s theorem has many connections to deep and fundamental results in discrete and computational geometry. For example, it generalizes Radon’s Lemma, one of the most important results about convexity. It also implies the centerpoint theorem and is related to the colorful Caratheodory theorem. It is further a crucial ingredient in proofs of the first selection lemma as well as the existence of weak ε-nets, two deep results that play an important role in discrete geometry, as witnessed for example by the celebrated proof of the (p,q)-Theorem by Alon and Kleitman. More generally, Tverberg’s theorem is intimately related to the study of geometric transversals, in particular Helly-type theorems and intersection patterns of convex sets, an area that has recently regained attention through its importance in the emerging field of topological data analysis. As such, it is not surprising that Tverberg’s theorem has also found applications in the theory of data science and machine learning.
Due to its importance in discrete and computational geometry, many variants of Tverberg’s theorem have been proposed and studied. Examples of such variants include colorful versions, and versions with tolerance, where the convex hulls should still intersect even after removing some of the points. Maybe most famously, the topological Tverberg conjecture was one of the main drivers in the development of topological methods in discrete geometry and has only been disproven a few years ago.
Despite a significant body of work, many fundamental questions about Tverberg’s theorem are still not well- understood. These questions are both of a combinatorial nature, e.g., understanding the structure of Tverberg partitions, but also computational, like understanding the computational complexity of finding Tverberg partitions. Our project intends to fill many of these gaps in our understanding of Tverberg’s theorem. In particular, by bringing together a team with diverse background ranging from algebraic topology to combinatorial optimization, and employing several novel ideas and approaches based on very recent advances and new viewpoints in the area, we will prove new structural theorems as well as develop more efficient algorithms and better complexity bounds around Tverberg’s theorem. Our work connects to several important conjectures in the field that have been open since more than 20 years, and we expect that our work will resolve some of them.
Contact: B. Gärtner
Duration: 4 years (from Feb 2026)
