Structural properties of distance-bounded phylogenetic reconciliation

TL;DR

The paper addresses time-feasible reconciliations between host and parasite phylogenies under a distance bound $d$ on host switches, introducing the graphs $D^\varphi$ and $G^\varphi$ to capture temporal constraints. It shows that for $d=3$ and $d=4$, strong acyclicity can be certified by forbidding a finite set of short cycles, enabling local cycle checks (three cycle types for $d=3$ and 110 cycles for $d=4$). While this does not resolve the overall complexity, it provides a structural, locality-based understanding of distance-bounded reconciliations and lays groundwork for potential polynomial-time checks under these constraints. The results thus offer a new lens on time-feasibility in cophylogenies and point to algorithmic avenues for constructing feasible reconciliations under distance bounds.

Abstract

Phylogenetic reconciliation seeks to explain host-symbiont co-evolution by mapping parasite trees onto host trees through events such as cospeciation, duplication, host switching, and loss. Finding an optimal reconciliation that ensures time feasibility is computationally hard when timing information is incomplete, and the complexity remains open when host switches are restricted by a fixed maximum distance $d$. While the case $d=2$ is known to be polynomial, larger values are unresolved. In this paper, we study the cases $d=3$ and $d=4$. We show that although arbitrarily large cycles may occur, it suffices to check only bounded-size cycles (we provide a complete list), provided the reconciliation satisfies acyclicity (i.e., time-feasibility) in a stronger sense. These results do not resolve the general complexity, but highlight structural properties that advance the understanding of distance-bounded reconciliations.

Figures (11)

• Figure 1: An example instance $(H,P,\sigma)$ of the reconciliation problem (or of the problem distance-bounded reconciliation$(4)$). Trees $H$ and $P$ are shown here; $\sigma$ maps upper case letters to lower case letters. A reconciliation $\varphi$ of $(H,P,\sigma)$ is given by $\varphi(r_P)=r_H$, $\varphi(u)=\varphi(v')=\varphi(w)=h_4$, $\varphi(v)=\varphi(x)=h_5$, and $\varphi(u')=h_2$. switching edges (w.r.t. $\varphi$) are drawn in gray.
• Figure 2: Directed graphs $D^\varphi$ and $G^\varphi$ for the reconciliation $\varphi$ of \ref{['fig:HPsigma eg']}. Added arcs are drawn in gray in this diagram for better visibility.
• Figure 3:
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Theorems & Definitions (20)

• Definition 1: Reconciliation, Events of a parasite node
• Definition 2
• Definition 3
• Definition 4: stolzer12Donati2015
• Definition 5: Acyclicity stolzer12Donati2015
• Definition 6
• Definition 7: Strong acyclicity
• Definition 8
• Definition 9: Arcs in $H^+_d$ for $d\in \{2,3,4\}$
• Lemma 1