Persistent Perfect Phylogeny

• Persistent perfect phylogeny is a binary model that permits a single gain (0→1) and a single loss (1→0) per character on each root-to-leaf path.
• It extends classical perfect phylogeny by incorporating back mutations, bridging the gap between strict and more general Dollo models.
• Developments include graph-reduction methods and polynomial-time algorithms, offering a robust framework for evolutionary and combinatorial analyses.

Persistent Perfect Phylogeny is a character-based phylogenetic model for binary data in which each character may be gained at most once and lost at most once, with the gain and loss occurring on the same root-to-leaf path. In the literature it is also called persistent phylogeny, binary perfect phylogeny with persistent characters, and Dollo-1. The model is a strict extension of classical perfect phylogeny, which allows only one $0 \to 1$ event and no loss, and a strict restriction of general Dollo models, which allow one gain and multiple losses. Its study has developed along several axes: formal equivalences with incomplete perfect phylogeny, graph-reduction characterizations, polynomial-time recognition algorithms, constrained and optimization-based variants, and a combinatorial theory of persistent characters on fixed trees (Bonizzoni et al., 2011, Bonizzoni et al., 2016, Bonizzoni et al., 24 Jul 2025).

1. Formal model and basic definitions

The standard input is an $n \times m$ binary matrix $M$, with rows representing species or taxa and columns representing binary characters. In the classical formulation, a persistent perfect phylogeny is a rooted tree $T$ whose root is labeled by the all-zero vector and whose edges encode character-state changes. For each character $c_j$, there are at most two state-changing edges on a single root-to-leaf path: first an edge labeled $c_j^{+}$, corresponding to $0 \to 1$, and later an edge labeled $c_j^{-}$, corresponding to $1 \to 0$. Each row of $M$ must label exactly one node of the tree. This is the formal relaxation introduced to permit a single back mutation while preserving a strong global tree constraint (Bonizzoni et al., 2011).

A useful generalization allows an active set $n \times m$0 at the root. In that formulation, the root state satisfies $n \times m$1 iff $n \times m$2, which is important for recursive decompositions because subproblems need not start from the all-zero state. Within this broader view, perfect phylogeny is the special case in which no negative edge $n \times m$3 ever appears, while persistent phylogeny remains the case of at most one gain and at most one loss per character (Bonizzoni et al., 2016).

This places Persistent Perfect Phylogeny between two classical extremes. Perfect phylogeny corresponds to Dollo-$n \times m$4: one gain and no loss. Persistent phylogeny corresponds to Dollo-$n \times m$5: one gain and at most one loss. General Dollo-$n \times m$6 allows one gain and at most $n \times m$7 losses, and unrestricted Dollo parsimony allows one gain and arbitrarily many losses. The persistent model is therefore a “middle ground” between perfect phylogeny and Dollo in both biological interpretation and algorithmic structure (Gusfield, 2015).

2. Extended matrices, completions, and red-black graphs

A central reformulation replaces each character $n \times m$8 by a pair of columns, written either $n \times m$9 or $M$0. If $M$1, then the pair is set to $M$2. If $M$3, the unconstrained formulation replaces it by $M$4, while constrained variants may force $M$5 for selected species-character pairs. A completion assigns each $M$6 either $M$7 or $M$8. The fundamental equivalence is that $M$9 admits a persistent phylogeny if and only if there exists a completion $T$0 of the extended matrix $T$1 such that $T$2 has a classical perfect phylogeny (Bonizzoni et al., 2011).

This completion view is encoded graph-theoretically by the red-black graph. Its vertices are species and characters. In the basic formulation, a black edge $T$3 corresponds to $T$4 and $T$5; in later constrained formulations, a red edge corresponds to $T$6. The graph evolves by realizing characters. Realizing an inactive character adds red edges to species in its connected component that do not currently have the character, removes all black edges incident to that character, and may isolate free or red-universal characters, whose incident edges are then deleted. The endpoint of a successful realization sequence is an edgeless graph (Bonizzoni et al., 2011, Bonizzoni et al., 24 Jul 2025).

The operative notion is a successful c-reduction, or equivalently a reduction ordering of characters. In the 2011 formulation, a successful reduction is an ordering of all characters whose successive realizations leave the red-black graph e-empty. In the later Dollo-1 graph-recognition formulation, a graph $T$7 has a Dollo-1 phylogeny if and only if there exists an ordering $T$8 of its characters such that the graph is reduced to an edgeless one. The forbidden configuration governing failure is the red $T$9-graph, an induced $c_j$0 in the red-black setting; reducibility is characterized by the existence of an ordering that avoids generating such a structure during the realization process (Bonizzoni et al., 2011, Bonizzoni et al., 24 Jul 2025).

3. Complexity and algorithmic development

The earliest exact algorithmic treatment for Persistent Perfect Phylogeny used the extended-matrix and red-black-graph formulation to search over character realizations. That method was exact, polynomial in the number of species, and exponential in the number of characters. The exponential dependence arose from branching over character orderings while pruning partial completions that already contained the forbidden matrix for perfect phylogeny (Bonizzoni et al., 2011).

A subsequent development settled the long-open complexity question. Persistent Phylogeny was shown to be solvable in polynomial time by converting the input matrix into a reducible red-black graph, extracting the maximal characters, constructing the Hasse diagram of the species partial order $c_j$1, and recursively realizing a safe source. The proof hinges on the fact that solutions for maximal reducible graphs have a highly restricted normal form: they are either line-trees or branch-trees, and the initial positive path corresponds to a safe chain in the Hasse diagram. The resulting algorithm computes a successful c-reduction and hence a persistent phylogeny in polynomial time (Bonizzoni et al., 2016).

This result sharply separates Dollo-1 from higher-loss variants. The same paper states that given a binary matrix, deciding whether there exists a Dollo-$c_j$2 tree compatible with the matrix is NP-complete for any fixed $c_j$3, while computing a Dollo-$c_j$4 tree corresponds to the Perfect Phylogeny decision problem, which admits a simple linear-time algorithm. Persistent phylogeny thus occupies an unusual intermediate position: more expressive than perfect phylogeny, but still polynomial-time solvable (Bonizzoni et al., 2016).

Later work revisited the problem from a direct graph-recognition perspective. For maximal bipartite species-character graphs, where no character’s species set is properly contained in another’s, a polynomial-time recognition algorithm was given using only graph properties. That algorithm exploits the persistence of connectedness through partial reductions, partitions species into $c_j$5 and $c_j$6, partitions inactive characters into $c_j$7, $c_j$8, and $c_j$9, and repeatedly realizes a safe character whose existence is guaranteed by structural theorems. This does not replace the general polynomial-time result; rather, it provides a refined recognition theory for a restricted but structurally important class of instances (Bonizzoni et al., 24 Jul 2025).

4. Constrained variants, galled trees, and integer programming

A natural extension is the Constrained Persistent Perfect Phylogeny problem, usually abbreviated CP-PP. Here the input is a pair $c_j^{+}$0, where $c_j^{+}$1 is a set of zero entries of $c_j^{+}$2. If $c_j^{+}$3, then character $c_j^{+}$4 is absent in species $c_j^{+}$5 and cannot be persistent for $c_j^{+}$6; equivalently, neither $c_j^{+}$7 nor any of its ancestors may carry $c_j^{+}$8 as a persistent event. In the extended matrix this forces $c_j^{+}$9 instead of $0 \to 1$0. The same red-black formalism applies, and solvability is again equivalent to the existence of a successful c-reduction (Bonizzoni et al., 2014).

The constrained theory yields a tractable subclass. When the conflict graph is empty, the authors prove that there is a polynomial-time algorithm for CP-PP. In particular, in the unconstrained case $0 \to 1$1, every such matrix admits a persistent perfect phylogeny. The algorithm builds the partial order $0 \to 1$2 iff $0 \to 1$3 for every species $0 \to 1$4, realizes maximal elements that lie in the same connected component of the red-black graph, and iterates until all characters are realized or no solution is possible. For the general constrained problem, the same paper gives a parameterized branch-and-bound algorithm whose parameter is the number of characters (Bonizzoni et al., 2014).

In parallel with these combinatorial developments, a different line of work framed persistent phylogeny as a completion problem suitable for integer linear programming. That approach constructs an extended matrix $0 \to 1$5 with two columns $0 \to 1$6 and $0 \to 1$7 per original column $0 \to 1$8, imposes twin-cell equalities $0 \to 1$9, and uses feasibility of the resulting MIDPP formulation to decide existence. The same paper emphasized a structural relationship with galled trees: if binary data $c_j^{-}$0 can be represented by a galled tree, then $c_j^{-}$1 can be represented by a persistent phylogeny, and the galled tree can be converted to a persistent phylogeny in linear time. The converse does not hold in general, so galled trees form a proper subfamily inside persistent phylogeny (Gusfield, 2015).

The optimization perspective was also empirically consequential. Using PERILP.pl with Gurobi 6.0, the ILP approach was reported to handle instances up to about 1000 taxa and 500 sites, with a six-minute limit not reached until around 400 taxa and 400 sites in the reported runs. The companion heuristic of first testing galled-tree representability via galledtree.pl was especially attractive because every tested instance was solved in under one second. Historically, this work presented the complexity status of persistent phylogeny as open; it remains important as a practical algorithmic study despite the later polynomial-time solution (Gusfield, 2015).

5. Persistent characters as a combinatorial object

A complementary literature studies not the reconstruction problem itself but the structure of persistent characters on a fixed rooted binary phylogenetic tree. In that setting, a binary character $c_j^{-}$2 is persistent if it can be realized by at most one $c_j^{-}$3 change followed by at most one $c_j^{-}$4 change. The central characterization uses the first phase of Fitch’s algorithm. If the parsimony score $c_j^{-}$5, then $c_j^{-}$6 is not persistent; if $c_j^{-}$7, then $c_j^{-}$8 is persistent; and if $c_j^{-}$9, persistence holds exactly when the two $1 \to 0$0-union nodes produced by Fitch’s first phase lie on a single ancestor-descendant chain and the induced state sets satisfy a specific nested condition. In this regime, a persistent character has a unique minimal persistent extension (Wicke et al., 2018).

That same work establishes a quantitative theory of persistence counts. Let $1 \to 0$1 denote the number of persistent binary characters on a rooted binary phylogenetic tree $1 \to 0$2, $1 \to 0$3 the number of persistent characters with parsimony score $1 \to 0$4, and $1 \to 0$5 the Sackin index. Then

$1 \to 0$6

Accordingly, the number of persistent characters depends only on tree balance. More imbalanced trees have more persistent characters; caterpillar trees maximize the count, and fully balanced trees minimize it when $1 \to 0$7 (Wicke et al., 2018).

A later Dollo-$1 \to 0$8 treatment situates persistent characters inside a broader combinatorial hierarchy. There, persistent characters are exactly the union of Dollo-0 and Dollo-1 characters. For a binary character $1 \to 0$9, the 1-tree $M$0 is the minimum induced subtree connecting all leaves in state $M$1, and the unique Dollo-$M$2 labeling assigns state $M$3 precisely to the vertices of $M$4. The Dollo score satisfies

$M$5

for characters with at least two 1-leaves, and a direct corollary is that $M$6 is persistent on $M$7 iff $M$8 contains at most two degree-2 nodes. The same paper shows that the close relationship between persistent characters and Fitch parsimony does not extend to general Dollo-$M$9 characters, and that the Sackin-index formula for persistence counts does not generalize beyond the persistent case (Bouckaert et al., 2020).

6. Relation to adjacent models, inference frameworks, and open contrasts

Persistent Perfect Phylogeny is best understood relative to the strict Perfect Phylogeny Model used in many inference systems for noisy mutation-frequency data. Under that baseline model, each genomic site mutates at most once and mutations are never lost. Exact solvers such as EXACT explore the full space of small perfect-phylogeny trees and compute exact likelihood-ranked tree distributions under noisy VAF data, while the projection algorithm based on Moreau’s decomposition gives an exact finite-step solver for the fixed-tree projection subproblem with $n \times m$00 time and $n \times m$01 memory (Ray et al., 2019, Jia et al., 2018). These methods do not solve persistent phylogeny directly, but they define a strict benchmark against which more permissive models can be evaluated.

This distinction matters because ambiguity is already substantial in the strict model. In the perfect phylogeny mixture setting, multiple trees can explain the same noiseless observation matrix $n \times m$02, and recent work showed that longitudinal constraints often fail to reduce the number of distinct trees that explain the observations, while alternative dynamic constraints can reduce ambiguity in a neighborhood of the true tree (Marangola et al., 31 Dec 2025). A plausible implication is that some empirical motivation for persistent models arises not only from biological losses, but also from degeneracy already present in strict perfect-phylogeny inference.

The broader perfect-phylogeny literature also clarifies what persistent phylogeny is not. For ordinary perfect phylogeny, the uniqueness problem for quartet-based tree reconstruction is NP-hard (Habib et al., 2010), and the local obstructions conjecture fails for 8-state characters: for every $n \times m$03, there exists a set of 8-state characters with no perfect phylogeny although every subset of at most $n \times m$04 characters has one (Iersel et al., 2018). These results concern stricter, nonpersistent models, but they show that even without losses the interaction between compatibility, uniqueness, and efficient certification is delicate. Persistent Perfect Phylogeny inherits that broader context while adding its own characteristic gain-once, lose-once structure.

In contemporary phylogenetics, Persistent Perfect Phylogeny therefore functions simultaneously as a biological relaxation, a graph-reduction problem, and a combinatorial class of characters. Its mature theory now includes exact reductions to perfect-phylogeny completion, polynomial-time recognition, constrained and optimization-based algorithms, connections to galled trees, and a detailed combinatorial characterization of persistent characters via Fitch sets, 1-trees, and the Sackin index (Bonizzoni et al., 2016, Gusfield, 2015, Wicke et al., 2018, Bouckaert et al., 2020).