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.
