An improved approximation ratio for the jump number problem on interval orders

The jump number problem is to find a linear extension of a given poset that minimizes the number of incomparable adjacent elements. The problem is NP-hard even in the class of interval orders and so three different 3/2-approximation algorithms for this case have been proposed respectively by Felsner, Sysło, and Mitas. We build upon the approach of Mitas, where the problem is reduced to packing vertex-disjoint edges and odd cycles in a certain graph. We prove that the requirement of independence between edges and cycles may be abandoned. In effect, the jump number problem is reduced to the set cover problem, and the 4/3-approximation algorithm of Duh and Fürer for the 3-set cover problem is used to show that the approximation ratio of the jump number problem on interval orders is tighter than 89/60.