CSC 5160 : Combinatorial Optimization and Approximation Algorithms

In this lecture, the focus is on submodular function in combinatorial optimizations. The first class of submodular functions which was studied thoroughly was the class of matroid rank functions. The flourishing stage of matroid theory came with Jack Edmonds’ work in 1960s, when he gave a minmax formula and an efficient algorithm to the matroid partition problem, from which the matroid intersection theorem and its algorithm can be derived. Generalizing certain basic properties of matroid polyhedra, Edmonds began the systematical study of submodularity. An important work along this line is Edmonds and Giles’ 1977 paper on submodular flow, which models a large class of useful and efficiently solvable combinatorial optimization problems. Although we will not introduce matroids in this course, we can still try to see how submodularity provides a unifying theme for many polynomial time solvable problems.