Lectures on Jump Systems Lectures on Jump Systems

This document contains notes that accompanied a series of four lectures on jump systems. These lectures were presented at the Center of Parallel Computing to an audience consisting mainly of graduate students. Abstract A jump system is a nonempty set of integral vectors that satisfy a certain exchange axiom. of Lovv asz. A degree system of a graph G is the set of degree sequences of all subgraphs of G. Degree systems are the primary example of jump systems. Other examples come from matroids and from two generalizations of matroids (polymatroids and delta{matroids). Discussion of these special cases will be kept to a minimum, and will only be used to motivate certain results. The main result is a min{max formula of Lovv asz for the distance of an integral point from a jump system. This formula generalizes two of the more important min{max theorems in combinatorial optimization; namely, Tutte's f{factor{theorem, and Edmonds' matroid intersection theorem. Other points of interest are the existence of a greedy algorithm for optimizing linear functions, and a characterization of the convex hulls of jump systems. Even apart from the possibility of obtaining very general theorems, jump systems are appealing due to their simple deenition and elegant structure.