Assume that the problem Qo is not solvable in polynomial time. For theories T containing a sufficiently rich part of true arithmetic we characterize T U {ConT} as the minimal extension of T proving for some algorithm that it decides Qo as fast as any algorithm B with the property that T proves that B decides Qo. Here, ConT claims the consistency of T. Moreover, we characterize problems with an optimal algorithm in terms of arithmetical theories.