 Approximation Algorithm Counterexample - Related to Set Cover Problem

Subject Computer Science Data Structures and Algorithms

Question

Give a counterexample to the following claim:
“A set cover instance in which each element is in exactly f sets has a (1 /f )-integral optimal fractional solution (i.e., in which each set is picked an integer multiple of 1 /f).”

Solution Preview

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Such an instance (of set cover problem) contains collection of at most k subsets Si included in U such that U is covered by their union. We can consider a simple example like below:
U={x,y,z}; Si={{x,y},{y,z},{z,x},{x,y,z}}...

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