Complexity of the Jump Number Problem

The literature concerning the jump number is vast. In what follows we mention some of the most relevant results regarding the complexity of this parameter.

Pulleyblank [139] has shown that determining the jump number of a general poset is NP-hard. On the other hand, there are efficient algorithms for many interesting classes of posets. Chein and Martin [29] give a simple algorithm to determine the jump number in forests. Habib [81] presents new algorithms for forests, for graphs having a unique cycle, and for graphs having only disjoint cycles. Cogis and Habib [35] introduce the notion of greedy linear extension. They prove that every such exten- sion is optimal for the jump number in series-parallel posets, obtaining an algorithm for this poset class. Rival [144] extends this result to N -free posets. Later, Gierz and Poguntke [69] show that the jump number is polynomial for cycle-series-parallel

posets.

Duffus et al. [45] have shown that for alternating-cycle-free posets, the jump num- ber of a poset is always one unit smaller than its width (see Theorem 4.1.1). Chein and Habib [28] give algorithms for posets of width two and later Colbourn and Pul- leyblank [36] provide a dynamic program algorithm for posets of bounded width. Steiner [155] also gives polynomial time algorithms for posets admitting a particular

bounded decomposition width.

Since the jump number is polynomial for posets of bounded width, it is natural to ask the same question for posets of bounded height. On the negative side, Pul- leyblank’s proof [139] of NP-hardness shows that the jump number is NP-hard even

when restricted to posets of height 2 or equivalently, to bipartite graphs. Müller [121] has extended this negative result to chordal bipartite graphs. On the positive side, there are efficient algorithms to compute the jump number of bipartite permutation graphs: an O(n + m)-time algorithm by Steiner and Stewart [156] and O(n)-time algorithms independently discovered by Fauck [53] and Brandstädt [16], where n and

m represent the numbers of vertices and edges of the graph respectively. The last

two algorithms require a succinct representation of the graphs (e.g. a permutation). There are also an O(n2_{)-time algorithm for biconvex graphs by Brandstädt [16], an}

O(m2_{)-time and an O(nm)-time algorithm for bipartite distance hereditary graphs}

developed by Müller [121] and Amilhastre et al. [2] respectively, and an O(n9_)-time

dynamic program algorithm by Dahlhaus [40] for convex graphs.

All the results in the previous paragraph are based on the equivalence for bipartite graphs between the jump number and the maximum alternating-cycle free matching problem (Theorem 4.1.2). For chordal bipartite graphs, the only induced cycles have length 4. Therefore, in this class, the jump number problem is equivalent to finding a maximum matching having no 4-alternating cycle. We study the latter problem in detail in Section 4.2. The algorithms for bipartite permutation and biconvex graphs are further inspired by a problem in computational geometry: the minimum rectangle cover of a biconvex rectangular region. We explore this problem in Section 4.3.1.

An important class of posets for which the complexity of this problem is still not settled is the class of two-dimensional posets. The tractability of the jump number problem on permutation graphs (that is, comparability graph of two-dimensional posets, see Lemma 3.4.3) is still open and conjectured to be polynomially solvable by Bouchitté and Habib [15]. As mentioned above, the jump number is polynomial for bipartite permutation graphs. Ceroi [23] extended this result to two-dimensional posets having bounded height. Interestingly enough, Ceroi [24] also gives a proof of the NP-hardness of a weighted version of this problem and conjectures, in contrast to Bouchitté and Habib, that the unweighted case is also NP-hard. Both results of Ceroi use a clever reduction to the maximum weight independent set of a particular family of rectangles. A similar construction for other types of graphs is studied in Section 5.2.

We end this survey by mentioning other results regarding the jump number. Mi- tas [117] has shown that the jump number problem is NP-hard also for interval posets. However, 3/2-approximation algorithms have been independently discovered by Mi- tas [117], Felsner [54] and Syslo [160]. As far as we know, these are the only ap- proximation results in this area, and it would be interesting to obtain approximation algorithms for other families of posets. It is also worth mentioning that, as shown by von Arnim and de la Higuera [3], this parameter is polynomial for the subclass of semiorders.

There has also been work on posets avoiding certain substructures (for exam- ple, N -free posets as stated before in this survey). Sharary and Zaguia [149] and Sharary [148] have shown that the jump number is polynomial for K-free posets and for Z-free posets respectively. Lozin and Gerber [106] propose some necessary condi- tions for the polynomial-solvability of the jump number on classes of bipartite graphs characterized by a finite family of forbidden induced graphs and give polynomial time

algorithms for some of these classes. In a different article, Lozin [105] shows that the jump number problem is also polynomial for E-free bipartite graphs.

Another interesting result is that the jump number is fixed parameter tractable: El-Zahar and Schmerl [51] have shown that the decision problem asking whether the jump number is at most a fixed number k is solvable in polynomial time and McCartin [114] has given an algorithm linear in the number of vertices (but factorial in k) to solve the same question.