Optimizing Phylogenetic Diversity with Ecological Constraints

Abstract

Abstract. Given an edge-weighted tree T with leaf set X, define the weight of a subset S of X as the sum of the edge-weights of the minimal subtree of T connecting the elements in S. It is known that the problem of selecting subsets of X of a given size to maximize this weight can be solved using a greedy algorithm. This optimization problem arises in conservation biology where the weight is referred to as the phylogenetic diversity of a taxa set S. Here, we consider the extension of this problem whereby we are only interested in selecting subsets of the taxa set that are ecologically 'viable'. Such subsets are specified by an acyclic digraph which represents, for example, a food web. This additional constraint makes the problem computationally hard. In this paper, we analyze the complexity of different variations of the extended problem.

Maximizing a Submodular Function with Viability Constraints

“…Consider the phylogenetic tree given in Figure 1 OptPDVC is known to be NP-hard [6], even for restricted classes of phylogenetic trees and dependency graphs.…”

“…S ← S ∪ {s} 5: end while i.e., as the minimum of the cardinality of the longest path that ends in a node without outgoing edges and k. If the food web is clear from the context we just write d instead of d(D). Note that the problem remains NP-hard for instances with d = 2, even if PD is additive [6]. Finally we define the costs c(A|B) of adding a set of species A to a set B…”

“…under viability constraints together with a cardinality constraint. We first review the greedy algorithm given by Faller et al [6] presented in Algorithm 1. The idea is that, in each step, one considers only species which either have no successors in the food web or for which one of the successors has already been selected (adding one of these species will keep the set viable).…”

“…Faller et al [6] show that the problem of maximizing phylogenetic diversity with viability constraints is NP-hard, even in simple special cases with constant depth (e.g. the food web is a directed tree of constant depth).…”

Maximizing a Submodular Function with Viability Constraints

“…Now we have that PD({A}) = 2, PD({B}) = 3 and PD({A, B}) = 4. OptPDVC is known to be NP-hard [6], even for restricted classes of phylogenetic trees and dependency graphs.…”