An algorithm for solving the jump number problem

Abstract

First, Cogis and Habib (RAIRO Inform. Théor. 13 (1979), 3–18) solved the jump number problem for series-parallel partially ordered sets (posets) by applying the greedy algorithm and then Rival (Proc. Amer. Math. Soc. 89 (1983), 387–394) extended their result to N-free posets. The author (Order 1 (1984), 7–19) provided an interpretation of the latter result in the terms of arc diagrams of posets explaining partly tractability of this special case.In this paper, we present an algorithm for solving the jump number problem on arbitrary posets which extends the author's approach applied to N-free posets and makes use of two new types of greedy chains in posets introduced in companion papers [8,9]. Complexity analysis of the algorithm supports our expectation that, going from N-free to arbitrary posets, the complexity of the problem increases with the number of dummy arcs required in their arc diagrams. The algorithm works in time which is linear in the poset size but factorial in the number of dummies, therefore it is a polynomial-time algorithm for posets with bounded number of dummies in their arc diagrams.