An algorithm for finding most likely explanations in valuation based systems

A method for finding a number of best explanations in so-called valuation based system is presented. Roughly speaking, the method allows to sort (decreasingly or increasingly) a function of many variables without explicit computation of values of this function. The only condition is that the function be decomposable, i.e. can be expressed as a combination of a number of low-dimensional functions called components. Two cases are considered: the combination operator has an inverse and a more elaborated case when the combination operator has no inverse.