Background: Orthologs inference is the starting point of most comparative genomics studies, and a plethora of methods have been designed in the last decade to address this challenging task. In this paper we focus on the problems of deciding consistency with a species tree (known or not) of a partial set of orthology/paralogy relationships C on a collection of n genes. Results: We give the first polynomial algorithm-more precisely a O(n 3) time algorithm-to decide whether C is consistent, even when the species tree is unknown. We also investigate a biologically meaningful optimization version of these problems, in which we wish to minimize the number of duplication events; unfortunately, we show that all these optimization problems are NP-hard and are unlikely to have good polynomial time approximation algorithms. Conclusions: Our polynomial algorithm for checking consistency has been implemented in Python and is available at https://github.com/UdeM-LBIT/OrthoPara-ConstraintChecker.