No Guarantee Unless P equals NP Benjamin Daggolu

Theorem 8.23 For each problem Π in NP, there is a polynomial-time map f from instances of Π to instances of Max3SAT and a fixed ε > 0 such that, for any instance x of Π, the following implications hold: X is a “yes” instance => opt(f(x))=|f(x)| X is a “no” instance => opt(f(x))<(1- ε )|f(x)|

here |f(x)| denotes the number of clauses in f(x) here |f(x)| denotes the number of clauses in f(x). f is a gap-creating reduction to Max3SAT. The theorem is limited to ruling out membership in PTAS and that we need other tools to rule out membership in Apx.

Theorem 8.25 Approximating the optimal solution to traveling salesman within any constant ratio is NP-hard. (pg-333)

Proof: We proof this theorem by contradiction Assumption: Consider an approximation algorithm A with absolute ratio RA= ε. We reuse the transformation from HC, but we produce large number tailored to the assumed ratio.

Given an instance of HC with n vertices, we can produce a instance of TSP with one city for each vertex and the distance between two cities is 1 where there exists an edge between the two corresponding vertices. This reduction produces enormous gap.

If an instance x of HC admits a solution, then the corresponding optimal tour uses only graph edges and thus has total length n. If x has no solution then the very best tour must move at least once between two cities not connected by an edge and thus has total length at least n-1+[n/ε].

The resultant gap exceeds the ratio ε which is a contradiction of our assumption. Therefore we can say that the general version of TSP is not in Apx, unlike its restriction to instances obeying the triangle inequality, for which a 2/3 approximation is known.

Thank you!