 
Summary: COMPLEXITY: Exercise No. 6
due in two weeks
1. (Test 99)Consider the following problem:
Input: Sets S 1 ; : : : ; S k ` N where jS i j = 3 for all i.
Goal: Find a set I ` f1; 2; : : : ; kg with maximum size such that S i `` S j = OE for all i; j 2 I.
Give a polynomial time approximation algorithm to this problem with a constant approxima
tion ratio.
2. (Test 99)Consider the MINIMUM STEINER TREE problem:
Input: A complete graph G = (V; E), a subset of the vertices X ` V , and a length function
l(e) ? 0 defined on the edges. The lengths satisfy the triangle inequality.
Goal: Find a tree T = (W; F ) such that X ` W ` V , F ` E and Length(T ) =
P
e2F
l(e) is
minimal.
Give an approximation algorithm for this problem whose approximation ratio is Ÿ 2.
Note: If X = V then the optimal tree is a minimum spanning tree, but this is not true if
X ae V .
3. Give an approximation algorithm for MAXIMUM CLIQUE whose inverse ratio is O(n= log n)
where n is the number of vertices in the input graph (i.e., there is a constant c such that
