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Discrete Mathematics: methods, challenges and applications

Final Report Summary - DMMCA (Discrete Mathematics: methods, challenges and applications)

Significant progress has been made during the project. In what follows we only briefly describe some of the highlights.

Settling a well studied open problem raised by Matousek, Seidel and Welzl in 1990, we showed that there is a set of points in the plane, so that any subset of it that contains at least one point of any line containing a significant number of points of the original set has to be substantially bigger than believed earlier by most researchers in the area. The proof is based on ideas in Ramsey Theory, proved by topological arguments. The paper with this result appeared in FOCS-one of the two main annual conferences in theoretical computer science, and its full version will appear in Discrete and Computational Geometry, the leading journal in this area.

In a paper with Fischer, Procaccia and Tennenholtz, that received the best paper award in TARK 2011, we applied combinatorial and probabilistic techniques to tackle a problem in Social Choice Theory. We showed that in a natural well-defined model, any deterministic algorithm can be manipulated, whereas randomized algorithms can do much better.

It is well known that one can color the plane by 7 colors with no monochromatic configuration consisting of the two endpoints of a unit segment. In sharp contrast we show that for any finite set of points K in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane, so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of K. The proof, obtained in joint work with Kostochka, combines combinatorial and probabilistic ideas.

Tobias Muller, a postdoc supported by the grant, considered in joint work with several researchers a certain natural model for generating a random geometric graph. Settling a question of Penrose, he identified the range of parameters under which the graph obtained is likely to contain a Hamilton cycle, that is, a cycle passing exactly once in each vertex.

The use of computational and mathematical modeling to investigate the behavior of biological systems has a long and fruitful history. In a paper with several coauthors published in Science we present an interesting reversal of this approach: using the behavior of a biological system to deduce a solution of a well studied computational problem for graphs.

Graph theoretic tools are helpful in the study of problems in Information Theory and Communication. In joint work with Moitra and Sudakov we construct dense graphs consisting of a union of large pairwise edge disjoint large induced matchings, and use it to design an efficient communication protocol. This settles a problem of Meshulam and improves results of Birk, Linial and Meshulam.