Transversal Graphs for Partially Ordered Sets: Sequencing, Merging and Scheduling Problems

Abstract

This paper introduces an approach to solving combinatorial optimization problems on partially ordered sets by the reduction to searching source-sink paths in the related transversal graphs. Different techniques are demonstrated in application to finding consistent supersequences, merging partially ordered sets, and machine scheduling with precedence constraints. Extending the approach to labeled partially ordered sets we also propose a solution for the smallest superplan problem and show its equivalence to the well studied coarsest regular refinement problem. For partially ordered sets of a fixed width the number of vertices in their transversal graphs is polynomial, so the reduction allows us easily to establish that many related problems are solvable in polynomial or pseudopolynomial time. For example, we establish that the longest consistent supersequence problem with a fixed number of given strings can be solved in polynomial time, and that the precedence-constrained release-date maximum- or total-cost preemptive or nonpreemptive job-shop scheduling problem with a fixed number of jobs can be solved in pseudopolynomial time. We also show that transversal graphs can be used to generalize and strengthen similar results obtained earlier by dynamic programming.