The jump number problem: Exact and parameterized

Abstract

The Jump Number problem asks to find a linear extension of a given partially ordered set that minimizes the total number of jumps, i.e., the total number of consecutive pairs of elements that are incomparable originally. The problem is known to be NP-complete even on posets of height one and on interval orders. It has also been shown to be fixed-parameter tractable. Finally, the Jump Number problem can be solved in time O*(2n) by dynamic programming. In this paper we present an exact algorithm to solve Jump Number in O(1.8638n) time. We also show that the Jump Number problem on interval orders can be solved by an O(1.7593n) time algorithm, and prove fixed-parameter tractability in terms of width w by an O*(2 w) time algorithm. Furthermore, we give an almost-linear kernel for Jump Number on interval orders for parameterization by the number of jumps. © 2013 Springer International Publishing.