A tabu search approach to the jump number problem

Abstract

We consider algorithmics for the jump numberproblem, which is to generate a linear extension of a given poset,minimizing the number of incomparable adjacent pairs. Since thisproblem is NP-hard on interval orders and open on two-dimensionalposets, approximation algorithms or fast exact algorithms are indemand.In this paper, succeeding from the work of the second namedauthor on semi-strongly greedy linear extensions, we develop ametaheuristic algorithm to approximate the jump number with thetabu search paradigm. To benchmark the proposed procedure, weinfer from the previous work of Mitas [Order 8 (1991), 115–132] anew fast exact algorithm for the case of interval orders, and from theresults of Ceroi [Order 20 (2003), 1–11] a lower bound for the jumpnumber of two-dimensional posets. Moreover, by other techniqueswe prove an approximation ratio ofn/log lognfor 2D orders.