Selecting inversions in a chord progression with the triad by a lexicographic bi-criteria optimization

Abstract

We are given a sequence of m machines and an ordered set of n jobs. The n jobs are served by the m-machine system in their indexing order. We are also given a collection of p_j machine subsets for each job j ∈ N, where N denotes the set of jobs. We are primarily asked to choose a machine subset from the p_j options for each job j ∈ N so that the similarity sum over a series of chosen n machine subsets is maximized, where a degree of similarity of two machine subsets is defined to be the number of common machines between the two machine subsets. The series of chosen n machine subsets also induces a subsequence of machines for serving the n jobs. We call the length of an induced machine subsequence its stretch. We are secondarily asked to choose a machine subset from the p_j options for each job j ∈ N so that the stretch of the induced machine subsequence is minimized. The lexicographic machine subset selection problem is motivated by a musical issue of selecting inversions in a chord progression with the triad. In this paper, we propose a polynomial time exact algorithm for the lexicographic machine subset selection problem. We also demonstrate the solutions on three examples of chord progressions with the triad, and on randomly generated instances.