Incomparable Pairs Research Articles

An Optimal Algorithm to Find the Jump Number of Partially Ordered Sets

The jump number of a partially ordered set (poset) P is the minimum number of incomparable adjacent pairs (jumps) in some linear extension of P. The problem of finding a linear extension of P with minimum number of jumps (jump number problem) is known to be NP-hard in general and, at the best of our knowledge, no exact algorithm for general posets has been developed. In this paper, we give examples of applications of this problem and propose for the general case a new heuristic algorithm and an exact algorithm. Performances of both algorithms are experimentally evaluated on a set of randomly generated test problems.

The number of partially ordered sets with more points than incomparable pairs

Let p kn denote the number of unlabeled posets with n points and k unrelated pairs. We show that for k< n, these numbers satisfy a recursion formula of the form P kn = Σ k j=0 c j p k − j, n− j−1 , where the coefficients c j can be computed if the numbers q jm of all ordinally indecomposable posets with m points and j unrelated pairs are known for m − 1⩽ j⩽ k. The crucial lemma for the proof states that q jm =0 for j< m− 1. From the recursion formula it follows that p kn is a polynomial of degree k in the variable n and that P kn≥ n−1 k with asymptotic equality for fixed k. For small values of k, we determine these polynomials explicitly. At the other end of the scale, we find that q n−1,n = 2 n−3 for n⩾3. Similar results are obtained for the number of labeled posets with a fixed linear extension and a given number of unrelated pairs.

Balance theorems for height-2 posets

We prove that every height-2 finite poset with three or more points has an incomparable pair {x, y} such that the proportion of all linear extensions of the poset in which x is less than y is between 1/3 and 2/3. A related result of Komlos says that the containment interval [1/3, 2/3] shrinks to [1/2, 1/2] in the limit as the width of height-2 posets becomes large. We conjecture that a poset denoted by V m + maximizes the containment interval for height-2 posets of width m+1.

Fractional dimension of partial orders

Given a partially ordered setP=(X, ≤), a collection of linear extensions {L 1,L 2,...,L r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL i (andy<x in someL j ). For a positive integerk, we call a multiset {L 1,L 2,...,L t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask→∞. We prove various results about the fractional dimension of a poset.

The density of infima in the recursively enumerable degrees

We show that every nontrivial interval in the recursively enumerable degrees contains an incomparable pair which have an infimum in the recursively enumerable degrees.

A 3/2-approximation algorithm for the jump number of interval orders

The jump number of a partial order P is the minimum number of incomparable adjacent pairs in some linear extension of P. The jump number problem is known to be NP-hard in general. However some particular classes of posets admit easy calculation of the jump number.

Interval orders without odd crowns are defect optimal

The defect of a (partial) order relationP is defined to be the rank of the kernel of the associated incidence matrix. Gierz and Poguntke [7] have shown that the defect provides a lower bound for the number of incomparable adjacent pairs in an arbitrary topological sorting ofP. We show that this bound is sharp for interval orders without odd crowns. Furthermore, an efficient algorithm for topological sortings of such orders is presented which achieves the bound. We finally exhibit a natural matroid structure associated with the optimal topological sortings under consideration, which permits to solve the weighted case.

The spread of a partial order

The deficiency of a partial order is the number of incomparable pairs. Another measure, the spread of a partial order is defined and its relation to the deficiency is established. In certain cases which arise naturally in the analysis of various sorting and selection algorithms where the partial order is only implicitly defined, the spread of the partial order is easier to estimate directly and provides a convenient way to estimate the deficiency.

An Inhomogeneity in the Structure of Karp Degrees

We show that there is a recursive nonzero Karp (polynomial time many-one) degree which is not supremum of a minimal pair, i.e. of an incomparable pair of degrees with infimum 0, the degree of polynomial time computable sets. By existence of minimal pairs, this implies that there are nonisomorphic initial segments of Karp degrees.

On minimal pairs of enumeration degrees

For sets of natural numbers A and B, A is enumeration reducible to B if there is some effective algorithm which when given any enumeration of B will produce an enumeration of A. Gutteridge [5] has shown that in the upper semilattice of the enumeration degrees there are no minimal degrees (see Cooper [3]), and in this paper we study those pairs of degrees with gib 0. Case [1] constructed a minimal pair. This minimal pair construction can be relativised to any gib, and following a suggestion of Jockusch we can also fix one of the degrees and still construct the pair. These methods yield an easier proof of Case's exact pair theorem for countable ideals. 0″ is an upper bound for the minimal pair constructed in §1, and in §2 we improve this bound to any Σ2-high Δ2 degree. In contrast to this we show that every low degree c bounds a degree a which is not in any minimal pair bounded by c. The structure of the co-r.e. e-degrees is isomorphic to that of the r.e. Turing degrees, and Gutteridge has constructed co-r.e. degrees which form a minimal pair in the e-degrees. In §3 we show that if a, b is any minimal pair of co-r.e. degrees such that a is low then a, b is a minimal pair in the e-degrees (and so Gutteridge's result follows). As a corollary of this we can embed any countable distributive lattice and the two nondistributive five-element lattices in the e-degrees below 0′. However the lowness assumption is necessary, as we also prove that there is a minimal pair of (high) r.e. degrees which is not a minimal pair in the e-degrees (under the isomorphism). In §4 we present more concise proofs of some unpublished work of Lagemann on bounding incomparable pairs and embedding partial orderings.As usual, {Wi}i ∈ ω is the standard listing of the recursively enumerable sets, Du is the finite set with canonical index u and {‹ m, n ›}m, n ∈ ω is a recursive, one-to-one coding of the pairs of numbers onto the numbers. Capital italic letters will be variables over sets of natural numbers, and lower case boldface letters from the beginning of the alphabet will vary over degrees.

A Vertebrate Fauna of Argyll and the Inner Hebrides

PERTINACITY in an endeavour to carry out the results of a fixed idea has almost always been regarded as a virtue, even when the principle involved has seemed to be hopelessly mistaken, and thus the adherents of the Stuart and other lost causes still find sympathisers at the present day; but when none can doubt the value of the idea, the pertinacity with which it is supported, provided that obstinacy is left out, becomes a virtue that in these practical days is not easily exaggerated. Such pertinacity is conspicuously exhibited by the authors of the book before us, Mr. Harvie-Brown and his worthy coadjutor, Mr. Buckley. This “Vertebrate Fauna of Argyll and the Inner Hebrides” is the fifth of a series of volumes, the inception of which is vastly creditable to its founder, the gentleman first named, and to all concerned in its production—even to the printer's devil and the binder's apprentice. Some of its predecessors have before received notice in these columns;1 but it has perhaps never been made clear to the readers of NATURE that this series of books is placing the zoology of the northern Kingdom on a footing which has not been attained, nor is likely to be attained in the southern part of the island, even though there exist particular English works—but this solely so far as ornithology is concerned—of merit superior to any one of the Scottish productions, the volume on Orkney, which is of remarkable excellence, being perhaps an exception. It is not difficult, however, to account to a considerable extent for this superiority: the proportion of persons with a taste for natural history to the general population being presumably the same in both parts of Great Britain, the enormously greater population of England would naturally furnish a larger number than Scotland is able to show. This is not said in derogation of the northern kingdom. It has always been rich in botanists; and, among zoologists, the single name of William Macgillivray is enough to cover it with renown. However much his merits, and especially his originality, may have been obscured or underrated in his life-time, he has already been recognised by those who have taken the trouble to inform themselves, and especially by American writers, as the most original British worker in regard to the vertebrate division of animals, since the incomparable pair—Willughby and Ray. But of Macgillivray this is not the place to speak particularly. On some other occasion we hope we may say more of him, a man whose work by some unhappy fate failed to impress his contemporaries, and whose posthumous volume was oppressed by princely patronage—well-meant but ill-advised. He had little or no experience of “Argyll and the Inner Hebrides,” and really does not now come into our story.1