After this class, you should be able to:
Example 1.5: The lower bound, of size of a maximal matching, is half the size of an optimal vertex cover for the following infinite family of instances. Consider the complete graph
Kn
, where n
is odd. The size of any maximal matching is (n-1)/2
, whereas the size of an optimal cover is n-1
.
1/2
algorithms for the following. (Acyclic subgraph) Given a directed graph G = (V,E)
, pick a maximum cardinality set of edges from E
so that the resulting subgraph is acyclic. [Hint: Arbitrarily number the vertices and pick the bigger of the two sets, the forward-going edges and the backward-going edges. What scheme are you using to upper bound OPT
?]
Hn
is O(log n)
.