Algorithm Design

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Chapter 10 Extending the Limits of Tractability
(10.2) If G = (V, E) has n nodes and a vertex cover of size k, then G has at
most k(n - 1) k IVI edges, then it has no k-node vertex cover
Else let e=(u,u) be au edge of G
Kecursively check if either of G-{u} or G-{u}
has a vertex cover of size k-I
If neither of them does, then G has no k-node vertex cover
10.1 Finding Small Vertex Covers
Else, one of them (say, G-{u}) has a (k- l)-node vertex cover T
In this case, TU[u} is a k-node vertex cover of G
Endif
Endif
~ Analyzing the Algorithm
Now we bound the running time of this algorithm. Intuitively, we are searching
a "tree of possibilities"; we can picture the recursive execution of the algorithm
as giving rise to a tree, in which each node corresponds to a different recursive
call. A node corresponding to a recursive call with parameter k has, as children,
two nodes corresponding to recursive calls with parameter k - 1. Thus the tree
has a total of at most 2 k+l nodes. In each recursive call, we spend O(kn) time.
Thus, we can prove the following.
(10.4) The running time o[ the Vertex Cover Algorithm on an n-node graph,
with parameter k, is O(2 k . kn).
We could also prove this by a recurrence as follows. If T(n, k) denotes the
running time on an n-node graph with parameter k, then T(., .) satisfies the
following recurrence, for some absolute constant c:
T(n, 1) _< cn,
T(n, k) < 2T(n, k - 1) + ckn.
By induction on k _> 1, it is easy to prove that T(n, k) < c. 2kkn. Indeed, if this
is true for k - 1, then
T(n, k) < 2T(n - 1, k - 1) + ckn
< 2c. 2/~-1(k - 1)n + ckn
= c. 2kkn -- c. 2kn + ckn