The Complexity of Computing Second Solutions

Abstract

We study the complexity of computing second solutions for NP search problems, i. e., given a problem instance x and a valid solution y, we have to find another valid solution y'.
Our main result shows that for typical NP decision problems, the complexity of computing second solutions is completely determined by the choice of the type of solution (i. e., the specific function problem), but independent of the underlying decision problem. More precisely, we show that for every X ∈ NP that is 1-paddable (a weak form of paddability), different choices of the type of solution lead to different second solution problems, which altogether have the same degree structure as the entire class of NP search problems (FNP). In fact, each degree of difficulty within FNP does occur as a second solution problem for X. This proves that typical NP decision problems have no intrinsic complexity w. r. t. the search for a second solution, but only the specification of the type of solution determines this complexity. This explains the empirical observation that the difficulty of computing second solutions strongly depends on the formulation of the problem.
Moreover, we show that the complexities of a search problem and its second solution variant are independent in the following sense: For all search problems A and B representing two degrees of difficulty, there exists a search problem C such that  
1) C is as difficult as A and 
2) computing second solutions for C is as difficult as B.