Periodic Reporting for period 1 - ERMiD (Effective Random Methods in Discrete Mathematics)
Berichtszeitraum: 2023-01-01 bis 2025-06-30
The main objective of this project is to pursue research at the meeting point of graph theory, geometry and complexity theory, searching for novel and practical applications whenever possible. The famous Turán problem asks for the largest number of edges in a graph that contains no copy of a prescribed subgraph. What happens if, instead, an exact number of copies are required? The problem has some intriguing connection to search theory, when we have a set of vertices and a hidden graph that is an unknown copy of a specified subgraph plus some isolated vertices. In a query a vertex pair can be asked, and the problem is to determine the minimal number of questions which allows determining the graph. It was known that all but Turán-number many edges must be asked; this result implies that among the answers all but 1-Turán number of answers must be “no”. Another widely studied problem is to find a single fake coin in a heap of coins by asking whether chosen subsets contain the fake coin. We have studied the cooperative version where many agents ask questions but not all of them learn the answer. We looked at various models and obtained strict bounds on the number of necessary questions. A similar question is testing monotone graph properties, that is, properties which are invariant when removing vertices and edges. Such a property can be checked to hold with high probability by looking at a single random subgraph of constant size. In many important graph properties this sample size, however, is astronomical even for moderate values of "high probability". In case of posets, however, this sample size turned out to be quite mild, and its approximate order is determined for many important special cases.
Federated optimization is a novel solution for scenarios where a common function is evaluated on a central server but the data are derived from separate clients. We have investigated the security and privacy aspects of such optimization methods, and devised a lightweight protocol for achieving maximal security. Secret sharing is one of the most investigated primitives in theoretical cryptography. New, effective constructions for generalized threshold schemes were found using tools from finite geometry. It is the first in its kind for arbitrary parameters, and yields significant improvement compared to the previous results. Finite submodular optimization yields an estimate on the best secret sharing protocols. We have initiated a new line of research connecting such discrete optimization and their continuous versions.
We also have looked at problems at the crossroad of geometry and extremal graph theory. Properties of planar graphs are well-understood, where nodes of the graphs are points, and edges are broken lines connecting nodes. There are other graph representations which arise naturally in planning robot movements when the vertices are represented by polygonal (broken) lines, and two such vertices are connected by an if these lines intersect. This other representation has many interesting and intriguing properties. In contrast to the four-color theorem, the chromatic number of these graphs can be arbitrarily large. Large chromatic graphs with additional special properties are constructed, keeping the broken lines as simple as possible.
Federated optimization is a novel solution for scenarios where a common function is evaluated on a central server but the data are derived from separate clients. We have investigated the security and privacy aspects of such optimization methods, and devised a lightweight protocol for achieving maximal security. Secret sharing is one of the most investigated primitives in theoretical cryptography. New, effective constructions for generalized threshold schemes were found using tools from finite geometry. It is the first in its kind for arbitrary parameters, and yields significant improvement compared to the previous results. Finite submodular optimization yields an estimate on the best secret sharing protocols. We have initiated a new line of research connecting such discrete optimization and their continuous versions.
We also have looked at problems at the crossroad of geometry and extremal graph theory. Properties of planar graphs are well-understood, where nodes of the graphs are points, and edges are broken lines connecting nodes. There are other graph representations which arise naturally in planning robot movements when the vertices are represented by polygonal (broken) lines, and two such vertices are connected by an if these lines intersect. This other representation has many interesting and intriguing properties. In contrast to the four-color theorem, the chromatic number of these graphs can be arbitrarily large. Large chromatic graphs with additional special properties are constructed, keeping the broken lines as simple as possible.
During this brief time period many significant results were achieved, such as solving long-standing open problems, advancing the frontiers of important research areas, and opening new, promising venues. Among them we pinpoint results which settle previously unsolved problems, some of which were open for many years.
A conjecture of Branko Grünbaum from 1972 states that in the plane any simple arrangement of n pairwise intersecting pseudo circles can have at most 2n-2 digons. Agarwal et al. proved it for arrangements which pairwise intersect and in which there is a common point surrounded by all pseudo circles. Felsner, Roch and Scheucher showed it to hold when every pair of pseudo circles creates a digon. This over 50-year-old conjecture has been settled in full generality for pairwise intersecting circles.
A paper by Gerbner et al. started the systematic study of the Turán-type extremal theory on graphs with a linear order on their edge-set. In contrast to the case of ordinal graphs when the extremal function is either linear or grows as some power of the vertex number, these edge-ordered graphs behave differently. A characterization of the case when the edge-ordered graph has almost linear extremal function was conjectured. This conjecture is settled in the affirmative.
The 0-1 matrix extremal question is the ordered version of the Turán-type extremal theory of vertex ordered bipartite graphs. The most important open problem in this area is to figure out how large the extremal function of acyclic patterns can be. Before our work the largest such known function was slightly above n times log(n). We improved this bound to arbitrary power of the logarithm; and showed that the extremal function of a small acyclic pattern can even be higher, refuting a more than 20 years old conjecture that all such extremal functions should be of the form n times polylog n.
A conjecture of Branko Grünbaum from 1972 states that in the plane any simple arrangement of n pairwise intersecting pseudo circles can have at most 2n-2 digons. Agarwal et al. proved it for arrangements which pairwise intersect and in which there is a common point surrounded by all pseudo circles. Felsner, Roch and Scheucher showed it to hold when every pair of pseudo circles creates a digon. This over 50-year-old conjecture has been settled in full generality for pairwise intersecting circles.
A paper by Gerbner et al. started the systematic study of the Turán-type extremal theory on graphs with a linear order on their edge-set. In contrast to the case of ordinal graphs when the extremal function is either linear or grows as some power of the vertex number, these edge-ordered graphs behave differently. A characterization of the case when the edge-ordered graph has almost linear extremal function was conjectured. This conjecture is settled in the affirmative.
The 0-1 matrix extremal question is the ordered version of the Turán-type extremal theory of vertex ordered bipartite graphs. The most important open problem in this area is to figure out how large the extremal function of acyclic patterns can be. Before our work the largest such known function was slightly above n times log(n). We improved this bound to arbitrary power of the logarithm; and showed that the extremal function of a small acyclic pattern can even be higher, refuting a more than 20 years old conjecture that all such extremal functions should be of the form n times polylog n.
Solving long-standing unsolved problems always requires new ideas. In one of our achievements it was a special graph associated with the circle arrangement. The graph is then embedded into the plane with edges as (possibly intersecting) straight segments. Discovering forbidden intersection patterns has led to the stated result as well as to interesting generalizations. Another idea was a structure-discovering argument. If the graph G does not contain a given forbidden subgraph, then G must contain a subgraph with “more structure”. As this reduction must stop somewhere, we get a bound on the size of G. The main challenge is to find what this "structure" is. In our case it is a tricky density-like condition, and the reduction uses probabilistic arguments.
The key novelty in attacking the measurability problem of treeing – a limiting object encompassing typical properties of huge trees – is applying Lovász’ result on graph flows which have neither sink nor source; these are the so-called circulations. In another application, to estimate the query complexity of Boolean functions we relied on decision trees, which are rooted binary trees where the vertices represent the queries, the edges are labeled with yes and no, and at the leaves contain the final answer. Using a sophisticated computation we estimated the total number of such decision trees which, after more calculations, lead to both new results and several additional consequences in discrepancy theory.
Submodular functions are analogues of convex functions that enjoy numerous applications. Their structural properties have been investigated extensively, and they have applications in such diverse areas as information inequalities, operational research, combinatorial optimization and social sciences, and have also found fundamental applications in game theory and machine learning. Extremal submodular functions have a unique minimal generating set. Our aim was to develop and implement computational geometry techniques which can enumerate this generating set for a six-element base set. Using the developed methods we succeeded in computing over 360 billion members, and also in giving a reasonable estimate for the total number.
The key novelty in attacking the measurability problem of treeing – a limiting object encompassing typical properties of huge trees – is applying Lovász’ result on graph flows which have neither sink nor source; these are the so-called circulations. In another application, to estimate the query complexity of Boolean functions we relied on decision trees, which are rooted binary trees where the vertices represent the queries, the edges are labeled with yes and no, and at the leaves contain the final answer. Using a sophisticated computation we estimated the total number of such decision trees which, after more calculations, lead to both new results and several additional consequences in discrepancy theory.
Submodular functions are analogues of convex functions that enjoy numerous applications. Their structural properties have been investigated extensively, and they have applications in such diverse areas as information inequalities, operational research, combinatorial optimization and social sciences, and have also found fundamental applications in game theory and machine learning. Extremal submodular functions have a unique minimal generating set. Our aim was to develop and implement computational geometry techniques which can enumerate this generating set for a six-element base set. Using the developed methods we succeeded in computing over 360 billion members, and also in giving a reasonable estimate for the total number.