Mining Posets From Linear Orders

Abstract

There has been much research on the combinatorial problem of generating the linear extensions of a given poset. This paper focuses on the reverse of that problem, where the input is a set of linear orders, and the goal is to construct a poset or set of posets that generates the input. Such a problem finds applications in computational neuroscience, systems biology, paleontology, and physical plant engineering. In this paper, two algorithms are presented for efficiently finding a single poset, if such a poset exists, whose linear extensions are exactly the same as the input set of linear orders. The variation of the problem where a minimum set of posets that cover the input is also explored. This variation is shown to be polynomially solvable for one class of simple posets (kite(2) posets) but NP-complete for a related class (hammock(2,2,2) posets).General Terms: Algorithms.

On Finding Two Posets that Cover Given Linear Orders

“…As mentioned earlier, there is already a polynomial-time solution for the 1-Poset Cover Problem [10]. We can determine in O(mn 2 ) time, where m is the number of linear orders over a base set of n elements, the poset that covers a given set of linear orders.…”

“…There are essentially two ways in which the problem can be restricted or constrained. The first is to consider only a specific number of posets, say k. The problem when k = 1, or the 1-Poset Cover Problem, is in P [10]. The computational complexity of the problem when k = 2, or the 2-Poset Cover Problem, is not yet known.…”

“…Evolutionary orderings are usually expressed using trees because evolution starts with an origin and branches out to descendants. Other classes of posets that have been studied are hammock posets and leveled posets [10]. The 2-Tree-Poset Cover Problem then is a contrained variation where the goal is to determine if there exist two tree-posets that cover exactly all of the given set of linear orders.…”

“…The first and outermost for-loop iterates in O(n 2 ). One dominating execution inside it is the call to GeneratePoset, which runs in O(mn 2 ) [10]. Another dominating part is the two inner for-loops, in Lines 16-20 or Lines 28-32, that iterates through all the pairwise combinations of subsets A and subsets of B.…”

The Poset Cover Problem

“…Bubley and Dyer [34] use a rapidly mixing Markov chain to generate a random linear extension of a finite poset, sampled almost uniformly. Problems in mining order information from databases of sequences (see, e.g., [16,17,35,36]) have an inverted character from that of many computational problems involving posets. Here, a problem instance is a set of linear orders of items from some universal set, while a solution is one or more posets that well explain, through their linear extensions, a significant number of the linear orders.…”