This research project has two main parts. The first focuses on applications of the differential equations method for the analysis of greedy algorithms and random graph processes. Recent work of the investigator and Alan Frieze gives a new technique for employing this method to prove good performance of randomized algorithms on random graphs without an explicit solution (or numerical approximation of the solution) of the associated system of differential equations (the need for such often complicates application of the method). The key new idea is to use combinatorial observations to identify an invariant set in the autonomous system of ordinary differential equations that, by a standard application of the method, governs the evolution of the algorithm. Problems to be addressed in this part of the project include Hamiltonicity, matchings and 3-colorability of sparse random graphs. The second part of this research project is motivated by the problem of determining the Shannon capacities of graphs. The Shannon capacity is a function of the independence numbers of the powers of the graph. The investigator proposes a novel technique, based on combinatorial stability, for attaining upper bounds on the independence numbers of powers of odd cycles.
This research is in the areas of probabilistic and extremal combinatorics. Probabilistic combinatorics is comprised of the study of random combinatorial structures, randomized algorithms and probabilistic existence proofs for combinatorics (i.e. the probabilistic method). Extremal combinatorics considers questions of the form: `How large (or small) can an object that lies in a particular discrete mathematical system and satisfies a certain condition be?' We are often concerned with the behavior of the size of the extremal objects as the size of the underlying system goes to infinity. Interest in both fields grew extensively during the twentieth century. Pioneers, such as Paul Erdos, discovered many beautiful properties of extremal and random discrete structures, posed vast arrays of fascinating questions in these areas and devised powerful methods for solving them even before the close connections between probabilistic and extremal combinatorics and various other fields (including theoretical computer science, statistical physics and information theory) were discovered. The Shannon capacities of graphs are a classic example of the close interaction between extremal combinatorics and the theory of communication. The Shannon capacity gives a measure of the optimal zero-error performance of a noisy communication channel and is centrally important in information theory. At the same time, it gives rise to natural questions in extremal combinatorics.
