Let S be a PQI preference structure on a finite set A, where the relations P, Q, I stand for ‘strict preference’, ‘weak preference’ and ‘indifference’, respectively. Two specific preference structures: PQI semi orders and PQI interval orders, have been considered and characterised recently in such a way that is possible to associate to each element of A an interval such that P holds when one interval is completely to the right of the other, I holds when one interval is included to the other and Q holds when one interval is to the right of the other, but they do have a non empty intersection (Q modelling the hesitation). While the detection of a PQI semiorder is straightforward, the case of the PQI interval order is more difficult as the theorem of existence consists in a second-order formula. The paper presents an algorithm for detecting a PQI interval order and demonstrates that it is backtracking free. This result leads to a matrix version of the algorithm which can be proved to be polynomial.
