1. LPs for MST
I forgot to return to the LP for maximum spanning tree we’d created in class, and show why it wasn’t an integral LP. To remind you, here was the LP:
Here is an example of a graph: the thin edges have weight , and the thick ones have weight .
The maximum weight spanning tree has weight . However, the LP solution given here (with on the thin edges, and on the thick ones) has . It also has , and you can check it satisfies the cut condition. (In fact, the main gadget that allows us to show this LP has an “integrality gap” is to assign to the edges of the thin triangle — much like for the naive non-bipartite matching LP.
One thing to note: take an undirected graph and replace each undirected edge by two directed edges of the same weight, pointing in opposite directions. Now the max-weight arborescence in this digraph has the same weight as the maximum spanning tree in the original graph. So an LP that looks pretty similar to the one above (namely ) on that specially constructed directed graph gives an arborescence that corresponds to an integral maximum spanning tree on the original undirected graph.
1.1. Counterexamples for Greedy
Also, here are two counterexamples: the left one for Ankit’s suggestion of running a Prim-like greedy algorithm (starting from the root) for min-cost arborescence. The right one we already saw in lecture: for a naive Kruskal-like greedy algorithm.
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