On unrooted and root-uncertain variants of several well-known phylogenetic network problems

The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an \emph{unrooted} phylogenetic network displays (i.e. embeds) an \emph{unrooted} phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the Tree Bisection and Reconnect (TBR) distance of the two unrooted trees. In the third part of the paper we consider the "root uncertain" variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.Comment: 28 pages, 8 Figure

A QUBO formulation for the Tree Containment problem

Phylogenetic (evolutionary) trees and networks are leaf-labeled graphs that are widely used to represent the evolutionary relationships between entities such as species, languages, cancer cells, and viruses. To reconstruct and analyze phylogenetic networks, the problem of deciding whether or not a given rooted phylogenetic network embeds a given rooted phylogenetic tree is of recurring interest. This problem, formally know as Tree Containment, is NP-complete in general and polynomial-time solvable for certain classes of phylogenetic networks. In this paper, we connect ideas from quantum computing and phylogenetics to present an efficient Quadratic Unconstrained Binary Optimization formulation for Tree Containment in the general setting. For an instance (N,T) of Tree Containment, where N is a phylogenetic network with n_N vertices and T is a phylogenetic tree with n_T vertices, the number of logical qubits that are required for our formulation is O(n_N n_T).Comment: final version accepted for publication in Theoretical Computer Scienc

On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems

International audienceThe hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To thisend we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the “root uncertain” variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees

Urban accupuncture: Architecture as a catalyst for environmental and water conservation in the context of the Kilimanjaro Informal Settlement

The following dissertation will attempt to establish an approach to dealing with the issue of waste contamination and water conservation in the natural and urban landscapes of the riverbed, its rivers' edges and its man-made peripheries. This research locates itself at the northern boundary of the city of Windhoek along a stretch of polluted riverbed in the Kilimanjaro Informal Settlement (KIS) where public environments are undefined, unhealthy and in many ways disconnected from the greater metropolitan areas. In the creation of an architectural approach 'urban acupuncture' will be explored in an attempt to create Architecture that has the potential to influence areas beyond its physical boundaries and which can re-establish and re-imagine the value of the river for its unseen influence in shaping the city as rapid urbanisation is taking place. In this section of the city, particular aspects of environmental degradation, water conservation and lack of basic infrastructure form a basis of inquiry to which an urban framework has been proposed. Drawing on theories of landscape urbanism, this urban framework acts to establish an alternative and more efficient infrastructural system which collects, stores, recycles and reuses wastewater for both drinking and irrigation purposes. Seen as the bi-product of this urban framework, the KIS Agricultural Learning Centre has been proposed which provides the necessary link between this infrastructural insertion and both the public and social constructs of the Kilimanjaro Informal Settlement

Placing problems from phylogenetics and (quantified) propositional logic in the polynomial hierarchy

In this thesis, we consider the complexity of decision problems from two different areas of research and place them in the polynomial hierarchy: phylogenetics and (quantified) propositional logic. In phylogenetics, researchers study the evolutionary relationships between species. The evolution of a particular gene can often be represented by a single phylogenetic tree. However, in order to model non-tree-like events on a species level such as hybridization and lateral gene transfer, phylogenetic networks are used. They can be considered as a structure that embeds a whole set of phylogenetic trees which is called the display set of the network. There are many interesting questions revolving around display sets and one is often interested in the computational complexity of the considered problems for particular classes of networks. In this thesis, we present our results for different questions related to the display sets of two networks and place the corresponding decision problems in the polynomial hierarchy. Another interesting question concerns the reconstruction of networks: given a set T of phylogenetic trees, can we construct a phylogenetic network with certain properties that embeds all trees in T? For a class of networks that satisfies certain temporal properties, Humphries et al. (2013) established a characterization for when this is possible based on the existence of a particular structure for T, a so-called cherry-picking sequence. We obtain several complexity results for the existence of such a sequence: Deciding the existence of a cherry-picking sequence turns out to be NP-complete for each non-trivial number (i.e., at least two) of given trees. Thereby, we settle the open question stated by Humphries et al. (2013) on the complexity for the case |T| = 2. On the positive side, we identify a special case that we place in the complexity class P by exploring connections to automata theory. Regarding propositional logic, we present our complexity results for the classical satisfiability problem (and variants resp. quantified generalizations thereof) and place the considered variants in the polynomial hierarchy. A common theme is to consider bounded variable appearances in combination with other restrictions such as monotonicity of the clauses or planarity of the incidence graph. This research was inspired by the conjecture that Monotone 3-SAT remains NP-complete if each variable appears at most five times which was stated in the scribe notes of a lecture held by Erik Demaine; we confirm this conjecture in an even more restricted setting where each variable appears exactly four times

The right to human dignity : a study of communal water and sanitation facilities for the peri-urban settlement of Inanda.

Master of Architecture. University of KwaZulu-Natal, Durban, 2015.Sustainable urban sanitation presents one of the many challenges towards service delivery and is directly related to poverty alleviation. Without appropriate social infrastructure such as water and sanitation - communities in the developing world can easily spiral into a decline. Water and appropriate sanitation, centre on community building and support communities to achieve high standards of health, equality, and good quality housing, good schools, safe, clean and friendly neighbourhoods. Without these social support infrastructure, peri-urban settlements struggle to become cohesive, and living communities with a sense of place, belonging or identity. In the developing world, communities without access to water and sanitation facilities suffer from a wide range of social problems and a platform for social discourse. Women are closely related to sanitation and water usage due to their social responsibilities at home and within their communities. Women tend to manage households and are the primary caregivers to children and extended family, also playing a nurturing role for the vulnerable, disabled and sick in the community. In South Africa, women living in rural and peri-urban areas face significant challenges. They live within a cycle of poverty, without appropriate access to a private toilets or to clean drinking water at the home. This research paper sets out to achieve an understanding of the daily living conditions communities face both spatially and programmatically, with a focus on women living in the peri-urban settlement of Inanda Durban. The objective set out tackles how architecture can be envisioned to meet dignified possibilities for and enrich the livelihoods of communities through the provision of appropriate and sustainable and suitable water and sanitation