Sample Problems in 18.997: Topics in Combinatorial Optimization
Even though 18.997 does not have regular problem sets, problems are occasionally mentioned in lecture for people to work on. Here is a sample of such questions:

Petersen’s theorem (Lecture 3) says that every cubic bridgeless graph contains a perfect matching. Show that for any even set T of vertices, a bridgeless cubic graph (V,E) contains a Tjoin of cardinality at most V/2 where a Tjoin is a subgraph with odd degree at vertices in T and even at vertices not in T.

Show that a graph is such that every edge is in a perfect matching if and only if it admits a bipartite ear decomposition, where a bipartite ear decomposition is one that starts from an even cycle and repeatedly adds an odd path between vertices of different colors.

Consider a strongly connected digraph D and a valid ordering O (lectures 7, 8). Show that a covering of the vertices by directed cycles of total minimum index can be obtained by network flows. What does network flow duality tells you here?

(Open.) Consider a matroid and assume S can be partitioned into k bases. Can one number cyclically be the element of S such that any sequence of k consecutive elements form a basis? Prove it when the numbering is not required to be cyclic.

Read a proof of (the strong version of) NashWilliams’s theorem saying that any undirected graph can be oriented so as to preserve half of the local connectivity rounded down. Can you provide an elementary proof?