VertexCoverByRounding

1. Let x be a min-cost fractional vertex cover.
2. Return the set S of vertices v having x(v) ≥ 1/2.

prove: The set S of vertices is a vertex cover. prove: The cost of S is at most twice the minimum cost of any vertex cover.

The ratio

(cost of optimal solution to original problem)/(cost of optimal solution to relaxed problem)

is called the integrality gap of the integer linear program. (In case of a maximization problem, the integrality gap is the reciprocal. The integrality gap of the problem is the worst-case integrality gap for any instance of the problem.

prove: The integrality gap of the above integer linear program for min-cost vertex cover is 2.