Computing an Evolutionary Ordering is Hard

Abstract

We study the problem of computing evolutionary orderings of families of sets, as introduced by Little and Campbell [1]. Definition 1. Let S be a family of subsets of some universe U . We say that S is evolutionary if there exists an ordering of its sets S = {S 1 , S 2 , . . . , S m } such that:• Each set brings a new element, i.e. S i i−1 j=1 S j• Each set, except the first one, has an old element, i.e. S i ∩ i−1 j=1 S j = ∅ The associated algorithmic problem is the following: Evolutionary Ordering Input: A family of subsets S of some universe U . Question: Is S evolutionary?We determine the computational complexity of this problem.Theorem 1. Evolutionary Ordering is NP-hard.

Self-generating autocatalytic networks: structural results, algorithms, and their relevance to evolutionary processes

“…A further question is the following: Given an element x in the RAF interior of 𝒬 and an element y in the RAF boundary, is there a viable (i.e. catalysed) path within the maxRAF from x to y (this problem can be easily solved by standard graph theoretic techniques, even though an ‘evolutionary’ ordering of an arbitrary collection of finite sets under a less constrained model is NP-complete [4]). Alternatively, one could ask, for two elements x, y in the RAF boundary, what common ancestor(s) in the RAF interior they could have had.…”

“…These reactions are written in a different way to the original description, however, it should be clear how to convert between them (e.g. {m1, m2} → {m1, m2, m3} becomes m1 + m2 → m1 + m2 + m3) 4. If viability is measured by an additive fitness function (i.e.…”