## Lecture #7 discussion (part II)

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:

$\displaystyle \begin{array}{rl} \max \sum_e & w_ex_e \\ \sum_e x_e &= |V| - 1 \\ x(\partial S) &\geq 1 \qquad \qquad \forall S \neq \emptyset, V \\ x &\geq 0 \end{array}$

Here is an example of a graph: the thin edges have weight ${0}$, and the thick ones have weight ${1}$.

The maximum weight spanning tree has weight ${2}$. However, the LP solution given here (with ${x_e = 1/2}$ on the thin edges, and ${x_e = 5/6}$ on the thick ones) has ${w^\top x = 3\cdot 5/6 = 2.5}$. It also has ${\sum_e x_e = 3\cdot 1/2 + 3\cdot 5/6 = |V| - 1}$, 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 ${1/2}$ 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 ${\max \{ w^\top x \mid x(\partial v) = 1, x(\partial S) \geq 1, x \geq 0 \}}$) 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.