STRONG CONDORCET CRITERION FOR THE LINEAR ORDERING PROBLEM

Abstract

The linear ordering problem is, given a nonnegative square matrix, to find a simultaneous permutation of the row and column indices such that the sum of the elements below the main diagonal of the permuted matrix is minimized. The linear ordering problem is an NP-hard optimization problem with a wide variety of applications, including triangulation of input-output tables and aggregation of individual preferences. In the context of the preference aggregation, Truchon (1998) showed that every optimal solution of a linear ordering problem satisfies a property called the extended Condorcet criterion, which states that there exists an ordered partition of the row and column index set of the given matrix such that the relative order of every pair of indices belonging to different components of the partition is fixed for all optimal solutions. Such fixing is desirable for investigating the whole set of the optimal solutions since we can reduce the computational time for enumerating them. In this study, we show that the optimal solutions of the linear ordering problem have an even stronger property than the extended Condorcet criterion, which enables us to have more pairs of indices whose relative orders are fixed for all optimal solutions.