AN O(N) ALGORITHM TO SOLVE THE BOTTLENECK TRAVELING SALESMAN PROBLEM RESTRICTED TO ORDERED PRODUCT MATRICES

The Bottleneck Traveling Salesman Problem (BTSP) is the problem of finding a Hamiltonian tour in a complete weighted digraph that minimizes the longest traveled distance between two successive vertices. The BTSP is studied in a graph where the distance matrix D = (d[i,j]) is given by d[i,j] = a[i] . b[j] with a [1] less-than-or-equal-to a[2] less-than-or-equal-to ... less-than-or-equal-to a[n] and b[1] greater-than-or-equal-to b[2] greater-than-or-equal-to ... greater-than-or-equal-to b[n]. It is observed that such so-called ordered product matrices (OPMs) have the following property. They are either ''doubly graded matrices'' or special ''max-distribution matrices''. Using this characterization, it is shown that there is an O(n) algorithm to solve the BTSP restricted to OPMs.