In a paper published in J. ACM in 1990, Tobias Nipkow asserted that the problemof deciding whether or not an equation over a nontrivial functionally complete algebra has a solution is NP-complete. However, close examination of the reduction used shows that only a weaker theorem follows from his proof, namely that deciding whether or not a system of equations has a solution is NP-complete over s...

We introduce a new class of NP problems called ANOTHER SOLUTION PROBLEMs. For a given NP problem X, ANOTHER SOLUTION PROBLEM for X (ASP for X) is to ask, for a given instance I for X and its solution, whether there is another solution for I. The di culty of ASP for X may not be immediate from the di culty of X. For example, for some NP-complete problems such as 3SAT or 3DM, it is easy to show t...

We use hypotheses of structural complexity theory to separate various NP-completeness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is T-complete but not P tt -complete. We provide fairly thorough analyses of the hypotheses that we introduce.

We introduce a simple technique to obtain reductions between optimization constraint satisfaction problems. The technique uses the probabilistic method to reduce the size of disjunc-tions. As a rst application, we prove the Max NP-completeness of Max 3Sat without using the PCP theorem (thus solving an open question posed in Khanna et al. (1994)). Successively, we show that the \planar" restrict...