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.

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