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A Branch and Bound Method for a Clique Partitioning Problem

We consider here the problem of the approximation of m symmetric relations defined on a same finite set X into a so-called median equivalence relation (see below and [1]), with in particular two special cases: the one for which the m symmetric relations are equivalence relations (Regnier’s problem [4]), and the one of the approximation of only one symmetric relation (m = 1) by an equivalence relation (Zahn’s problem [6]). These problems arise for instance from the field of classification or clustering: in this case, X is a set of entities (which can be objects, people, projects, propositions, alternatives, and so on) that we want to gather in subsets of X in such a way that the elements of any such subset can be considered as similar while the objects of different subsets can be considered as dissimilar. Each symmetric relation is associated with a criterion specifying, for any pair {x, y} of entities, whether x and y are similar or not. Then we try to find the best compromise between all these criteria. This leads us, in Section 2, to state this problem as a graph theoretical problem, that we call CPP for clique partitioning problem. As this problem is NP-hard, we design in Section 3 a branch and bound algorithm to solve this problem, based on a Lagrangean relaxation method for the evaluation function.