Computing the jth solution of a first-order query

We design algorithms of “optimal" data complexity for several natural problems about first-order queries on structures of bounded degree. For that purpose, we first introduce a framework to deal with logical or combinatorial problems R ⊂ I x O whose instances x ∈ I may admit of several solutions R(x) = {y ∈ O : (x,y) ∈ R}. One associates to such a problem several specific tasks: compute a random (for the uniform probability distribution) solution y ∈ R(x); enumerate without repetition each solution yj in some specific linear order y0 < y1 < ... < yn-1 where R(x) = {y0,...,yn-1}; compute the solution yj of rank j in the linear order