Periodic Reporting for period 4 - CombiCompGeom (Combinatorial Aspects of Computational Geometry)
Période du rapport: 2021-03-01 au 2022-08-31
The project addresses long-standing combinatorial problems at the heart of Computational Geometry.
Computational Geometry is a branch of Theoretical Computer Science dedicated to solving basic geometric problems using effective algorithms.
Its practical importance lies in its numerous applications to mathematical programming, computer graphics, robotic motion planning, geographic information systems, computational biology, and many other areas.
Combinatorial or Discrete Geometry is an established and fascinating branch of pure mathematics which studies the combinatorial aspects of discrete geometric structures. It bears strong relations to such well-established mathematical areas
as algebraic topology, algebraic geometry, probability theory, graph theory and extremal combinatorics.
The most fundamental questions in Computational Geometry are expressed in the terms of arrangements, Voronoi diagrams, polytopes, intersection graphs, and other abstract structures from combinatorial geometry.
Thus, design and analysis of efficient geometric algorithms is (typically) based on understanding the combinatorial complexity of these key objects studied in Discrete Geometry, and on being able to manipulate them efficiently.
The project 'Combinatorial Aspects of Computational Geometry' explores the fruitful interplay between the two areas by addressing outstanding combinatorial questions at the heart of computational geometry.
Moreover, it demonstrates that some classical problems in combinatorial geometry can be successfully tackled using approaches that emerged within computational geometry.
Thus, project strengthens the existing bridges between the two academic communities.
The project has resulted in breakthroughs on fundamental problems that defied progress for the last 2-3 decades, including 1. improved upper bounds for weak epsilon-nets in all dimensions,
2. finding almost-linear-size families of pairwise crossing edges in dense geometric graphs, 3. a novel Crossing Lemma for contact graphs of Jordan curves which confirmed a 20-year-old conjecture of Richter and Thomassen.
Computational Geometry is a branch of Theoretical Computer Science dedicated to solving basic geometric problems using effective algorithms.
Its practical importance lies in its numerous applications to mathematical programming, computer graphics, robotic motion planning, geographic information systems, computational biology, and many other areas.
Combinatorial or Discrete Geometry is an established and fascinating branch of pure mathematics which studies the combinatorial aspects of discrete geometric structures. It bears strong relations to such well-established mathematical areas
as algebraic topology, algebraic geometry, probability theory, graph theory and extremal combinatorics.
The most fundamental questions in Computational Geometry are expressed in the terms of arrangements, Voronoi diagrams, polytopes, intersection graphs, and other abstract structures from combinatorial geometry.
Thus, design and analysis of efficient geometric algorithms is (typically) based on understanding the combinatorial complexity of these key objects studied in Discrete Geometry, and on being able to manipulate them efficiently.
The project 'Combinatorial Aspects of Computational Geometry' explores the fruitful interplay between the two areas by addressing outstanding combinatorial questions at the heart of computational geometry.
Moreover, it demonstrates that some classical problems in combinatorial geometry can be successfully tackled using approaches that emerged within computational geometry.
Thus, project strengthens the existing bridges between the two academic communities.
The project has resulted in breakthroughs on fundamental problems that defied progress for the last 2-3 decades, including 1. improved upper bounds for weak epsilon-nets in all dimensions,
2. finding almost-linear-size families of pairwise crossing edges in dense geometric graphs, 3. a novel Crossing Lemma for contact graphs of Jordan curves which confirmed a 20-year-old conjecture of Richter and Thomassen.
Several breakthrough results can be attributed to the project:
1. The PI has achieved the first progress in 25 years towards understanding the true asymptotic growth rates of weak epsilon-nets with respect to convex sets in the Euclidean plane.
The study of epsilon-nets is central to computational geometry, as they determine the performance of the best known algorithms for geometric hitting set/set-cover problems, and are used in most divide-and-conquer schemes.
It is a decades-old open problem to determine whether the growth rates of (optimal) weak-epsilon nets with respect to convex sets in the Euclidean space overly resemble those of strong epsilon-nets.
We achieve the first progress in 25 years by improving the 1992 planar bound of Alon, Barany, Furedi and Kleitman.
A series of 2 papers establishes improved bounds for weak epsilon-nets in all dimensions. It is based upon a curious reduction of the problem to incidence bounds between points and generalized combinatorial hyperplanes.
2. The PI's collaboration with J. Pach and G. Tardos shows that any finite set of points in the Euclidean plane determines almost linearly many pairwise crossing segments, and the result extends to all sufficiently dense geometric graphs.
Notice that the cornerstone Crossing Lemma (1982) in combinatorial geometry yields a tight relation between the number of edges in a geometric graph and the number of edge crossings in its planar embedding. The result bears enormous importance to computational geometry.
Unfortunately, the Crossing Lemma tells very little, if at all, of the underlying intersection structure (i.e. intersection graph) of the edges, and very little progress has been achieved in the passed decades.
Our lower bound on pairwise crossing edges is almost tight, and improves a much weaker 1990 lower bound of Aronov, Erdos, Goddard, Klugerman, Kleitman, Pach and Schulman.
3. Another collaboration of the PI with J. Pach and G. Tardos establishes an analogue of the Crossing Lemma for contact graphs of Jordan curves in the plane. That is, we establish a non-trivial relation between the number of touchings (i.e. points of tangency which occur if the two involved curves intersect only once) and the overall number of their intersection points in a family of Jordan curves. In particular, we establish the 1995 Richter-Thomassen Conjecture for families of closed Jordan curves.
In addition, a joint work of the PI with the project postdoc Leonardo Martínez-Sandoval, and the team visitor Edgardo Roldán-Pensado resulted in a a novel Colourful Helly Theorem which offers a complementary relation between transversals of various dimensions to a family of convex sets that satisfies Lovasz's Colourful Helly Hypothesis. As a by-product, this study introduced to the discrete geometry community an intriguing conjecture on lines that cross a family of pairwise intersecting convex sets in the Euclidean 3-space.
1. The PI has achieved the first progress in 25 years towards understanding the true asymptotic growth rates of weak epsilon-nets with respect to convex sets in the Euclidean plane.
The study of epsilon-nets is central to computational geometry, as they determine the performance of the best known algorithms for geometric hitting set/set-cover problems, and are used in most divide-and-conquer schemes.
It is a decades-old open problem to determine whether the growth rates of (optimal) weak-epsilon nets with respect to convex sets in the Euclidean space overly resemble those of strong epsilon-nets.
We achieve the first progress in 25 years by improving the 1992 planar bound of Alon, Barany, Furedi and Kleitman.
A series of 2 papers establishes improved bounds for weak epsilon-nets in all dimensions. It is based upon a curious reduction of the problem to incidence bounds between points and generalized combinatorial hyperplanes.
2. The PI's collaboration with J. Pach and G. Tardos shows that any finite set of points in the Euclidean plane determines almost linearly many pairwise crossing segments, and the result extends to all sufficiently dense geometric graphs.
Notice that the cornerstone Crossing Lemma (1982) in combinatorial geometry yields a tight relation between the number of edges in a geometric graph and the number of edge crossings in its planar embedding. The result bears enormous importance to computational geometry.
Unfortunately, the Crossing Lemma tells very little, if at all, of the underlying intersection structure (i.e. intersection graph) of the edges, and very little progress has been achieved in the passed decades.
Our lower bound on pairwise crossing edges is almost tight, and improves a much weaker 1990 lower bound of Aronov, Erdos, Goddard, Klugerman, Kleitman, Pach and Schulman.
3. Another collaboration of the PI with J. Pach and G. Tardos establishes an analogue of the Crossing Lemma for contact graphs of Jordan curves in the plane. That is, we establish a non-trivial relation between the number of touchings (i.e. points of tangency which occur if the two involved curves intersect only once) and the overall number of their intersection points in a family of Jordan curves. In particular, we establish the 1995 Richter-Thomassen Conjecture for families of closed Jordan curves.
In addition, a joint work of the PI with the project postdoc Leonardo Martínez-Sandoval, and the team visitor Edgardo Roldán-Pensado resulted in a a novel Colourful Helly Theorem which offers a complementary relation between transversals of various dimensions to a family of convex sets that satisfies Lovasz's Colourful Helly Hypothesis. As a by-product, this study introduced to the discrete geometry community an intriguing conjecture on lines that cross a family of pairwise intersecting convex sets in the Euclidean 3-space.
These results constitute a major progress of the state-of-the-art in discrete and combinatorial geometry and indicate novel relations between fundamental combinatorial structures, thereby significantly changing the landscape of this area. This demonstrates the potential of Computer Science methods in solving pure mathematical questions in extremal combinatorics and combinatorial geometry in particular.
In the coming years, the used techniques will likely yield to further progress on such related problems and selections theorems (piercing simplices in a geometric hypergraph) and counting k-sets in higher dimensions, higher-dimensional variants of the Erdős Happy End problem, and far stronger bounds on edge numbers in k-quasi-planar geometric graphs.
In the coming years, the used techniques will likely yield to further progress on such related problems and selections theorems (piercing simplices in a geometric hypergraph) and counting k-sets in higher dimensions, higher-dimensional variants of the Erdős Happy End problem, and far stronger bounds on edge numbers in k-quasi-planar geometric graphs.