The mim-width of a graph is a powerful structural parameter that, when bounded by a constant, allows several hard problems to be polynomial-time solvable - with a recent meta-theorem encompassing a large class of problems [SODA2023]. Since its introduction, several variants such as sim-width and omim-width were developed, along with a linear version of these parameters. It was recently shown that mim-width and all these variants are all paraNP-hard, a consequence of the NP-hardness of distinguishing between graphs of linear mim-width at most 1211 and graphs of sim-width at least 1216 [ICALP2025]. The complexity of recognizing graphs of small width, particularly those close to 1, remained open, despite their especially attractive algorithmic applications. In this work, we show that the width recognition problems remain NP-hard even on small widths. Specifically, after introducing the novel parameter Omim-width sandwiched between omim-width and mim-width, we show that: (1) deciding whether a graph has sim-width = 1, omim-width = 1, or Omim-width = 1 is NP-hard, and the same is true for their linear variants; (2) the problems of deciding whether mim-width ≤ 2 or linear mim-width ≤ 2 are both NP-hard. Interestingly, our reductions are relatively simple and are from the Unrooted Quartet Consistency problem, which is of great interest in computational biology but is not commonly used in the theory of algorithms.

Leaf powers and pairwise compatibility graphs were introduced over twenty years ago as simplified graph models for phylogenetic trees. Despite significant research, several properties of these graph classes poorly understood. In this paper, we establish that the recognition problem for both classes is NP-complete. We extend this hardness result to a broader hierarchy of graph classes, including pairwise compatibility graphs and their generalizations, multi-interval pairwise compatibility graphs.

We present an output-sensitive algorithm for constructing incompatibility graphs between pairs of rooted or unrooted phylogenetic trees, in which edges represent incompatible clusters or splits. Incompatibility graphs capture conflicting evolutionary signals and play an important role in applications such as phylogenetic network reconstruction, supertree inference, and BHV distance computation. Existing approaches typically require O(n³/w) time using bitset operations, where n is the number of taxa and w is the machine word size. We introduce a new algorithm based on lowest common ancestor mappings that constructs the incompatibility graph of two rooted trees in optimal O(n+d) time, where d is the number of incompatibility edges. The method is extended to unrooted trees and trees with different leaf sets while preserving the same complexity, while also being relatively simple to implement. Experimental results on random and simulated phylogenetic trees show substantial practical speedups over existing implementations, particularly on sparse incompatibility graphs, which commonly arise in large datasets.

A graph $G=(V,E)$ is a $k$-leaf power if there is a tree $T$ whose leaves are the vertices of $G$ with the property that a pair of leaves $u$ and $v$ induce an edge in $G$ if and only if they are distance at most $k$ apart in $T$. For $k\le 4$, it is known that there exists a finite set $F_k$ of graphs such that the class $L(k)$ of $k$-leaf power graphs is characterized as the set of strongly chordal graphs that do not contain any graph in $F_k$ as an induced subgraph. We prove no such characterization holds for $k\ge 5$. That is, for any $k\ge 5$, there is no finite set $F_k$ of graphs such that $L(k)$ is equivalent to the set of strongly chordal graphs that do not contain as an induced subgraph any graph in $F_k$.

In the Cluster Editing problem, sometimes known as (unweighted) Correlation Clustering, we must insert and delete a minimum number of edges to achieve a graph in which every connected component is a clique. Owing to its applications in computational biology, social network analysis, machine learning, and others, this problem has been widely studied for decades and is still undergoing active research. There exist several parameterized algorithms for general graphs, but little is known about the complexity of the problem on specific classes of graphs. Among the few important results in this direction, if only deletions are allowed, the problem can be solved in polynomial time on cographs, which are the P₄-free graphs. However, the complexity of the broader editing problem on cographs is still open. We show that even on a very restricted subclass of cographs, the problem is NP-hard, W[1]-hard when parameterized by the number p of desired clusters, and that time n^o(p/log p) is forbidden under the ETH. This shows that the editing variant is substantially harder than the deletion-only case, and that hardness holds for the many superclasses of cographs (including graphs of clique-width at most 2, perfect graphs, circle graphs, permutation graphs). On the other hand, we provide an almost tight upper bound of time n^O(p), which is a consequence of a more general n^O(cw⋅p) time algorithm, where cw is the clique-width. Given that forbidding P₄s maintains NP-hardness, we look at {P₄, C₄}-free graphs, also known as trivially perfect graphs, and provide a cubic-time algorithm for this class.

In this work, we study the problem of computing a maximum common contraction of two vertex-labeled graphs, i.e. how to make them identical by contracting as little edges as possible in the two graphs. We study the problem from a parameterized complexity point of view, using parameters such as the maximum degree, the degeneracy, the clique-width or treewidth of the input graphs as well as the number of allowed contractions. We put this complexity in perspective with that of the labeled contractibility problem, i.e determining whether a labeled graph is a contraction of another. Surprisingly, our results indicate very little difference between these problems in terms of parameterized complexity status. We only prove their status to differ when parameterizing by both the degeneracy and the number of allowed contractions, showing W[1]-hardness of the maximum common contraction problem in this case, whereas the contractibility problem is FPT.

One of the most studied problem in theoretical computer science, Vertex Cover , has been recently considered in the temporal graph framework. Here we study a Vertex Cover variant, called k- TimelineCover . Given a temporal graph k- TimelineCover asks to define an interval for each vertex so that for every temporal edge existing in a timestamp t , at least one of the endpoints has an interval that includes t . The goal is to decide whether it is possible to cover every temporal edge while using vertex intervals of total span at most k . k- TimelineCover has been shown to be NP-hard, but its parameterized complexity has not been fully understood when parameterizing by the span of the solution. We settle this open problem by giving an FPT algorithm that combines two techniques, a modified form of iterative compression and a reduction to Digraph Pair Cut .

In this contribution we study the Arborescence Reconfiguration on temporal digraphs ( Temporal Arborescence Reconfiguration ). The problem, given two temporal arborescences in a temporal digraph, asks for the minimum number of arc flips, i.e., arc exchanges, that result in a sequence of temporal arborescences transforming one into the other. We analyze the complexity of the problem, taking into account also its approximation and parameterized complexity, even in restricted cases. First, we solve an open problem showing that Temporal Arborescence Reconfiguration is NP-hard for two timestamps. Then we show that even if the two temporal arborescences differ only by two pairs of arcs, then the problem is not approximable within factor b ln ⁡ | V ( D ) | , for any constant 0 < b < 1 , where V ( D ) is the set of vertices of the temporal arborescences. Finally, we prove that Temporal Arborescence Reconfiguration is W[1]-hard when parameterized by the number of arc flips needed to transform one temporal arborescence into the other.

In phylogenetics , evolution is traditionally represented in a tree-like manner. However, phylogenetic networks can be more appropriate for representing evolutionary events such as hybridization, horizontal gene transfer, and others. In particular, the class of forest-based networks was recently introduced to represent introgression , in which genes are swapped between species. A network is forest-based if it can be obtained by adding arcs to a collection of trees, so that the endpoints of the new arcs are in different trees. This contrasts with so-called tree-based networks, which are formed by adding arcs within a single tree . We are interested in the computational complexity of recognizing forest-based networks, which was recently left as an open problem by Huber et al. It has been observed that forest-based networks coincide with directed acyclic graphs that can be partitioned into induced paths, each ending at a leaf of the original graph. Several types of path partitions have been studied in the graph theory literature, but to our best knowledge this type of ‘leaf induced path partition’ has not been directly considered before. The study of forest-based networks in terms of these partitions allows us to establish closer relationships between phylogenetics and algorithmic graph theory, and to provide answers to problems in both fields. More specifically, we show that deciding whether a network is forest-based is NP-complete, even on input networks that are tree-based, binary, and have only three leaves. This shows that partitioning a directed acyclic graph into a constant number of induced paths is NP-complete, answering a recent question of Fernau et al. We then show that the problem is polynomial-time solvable on binary networks with two leaves and on the recently introduced class of orchards, which we show to be always forest-based. Finally, for undirected graphs, we introduce unrooted forest-based networks and provide hardness results for this class as well.

In this paper, we lay the groundwork on the comparison of phylogenetic networks based on edge contractions and expansions as edit operations, as originally proposed by Robinson and Foulds to compare trees. We prove that these operations connect the space of all phylogenetic networks on the same set of leaves, even if we forbid contractions that create cycles. This allows to define an operational distance on this space, as the minimum number of contractions and expansions required to transform one network into another. We highlight the difference between this distance and the computation of the maximum common contraction between two networks. Given its ability to outline a common structure between them, which can provide valuable biological insights, we study the algorithmic aspects of the latter. We first prove that computing a maximum common contraction between two networks is NP-hard, even when the maximum degree, the size of the common contraction, or the number of leaves is bounded. We also provide lower bounds to the problem based on the Exponential-Time Hypothesis. Nonetheless, we do provide a polynomial-time algorithm for weakly-galled trees, a generalization of galled trees.

In this study, we investigate the problem of comparing gene trees reconciled with the same species tree using a novel semi-metric, called the Path-Label Reconciliation (PLR) dissimilarity measure. This approach not only quantifies differences in the topology of reconciled gene trees, but also considers discrepancies in predicted ancestral gene-species maps and speciation/duplication events, offering a refinement of existing metrics such as Robinson-Foulds (RF) and their labeled extensions LRF and ELRF. A tunable parameter α also allows users to adjust the balance between its species map and event labeling components. We show that PLR can be computed in linear time and that it is a semi-metric. We also discuss the diameters of reconciled gene tree measures, which are important in practice for normalization, and provide initial bounds on PLR, LRF, and ELRF. To validate PLR, we simulate reconciliations and perform comparisons with LRF and ELRF. The results show that PLR provides a more evenly distributed range of distances, making it less susceptible to overestimating differences in the presence of small topological changes, while at the same time being computationally efficient. Our findings suggest that the theoretical diameter is rarely reached in practice. The PLR measure advances phylogenetic reconciliation by combining theoretical rigor with practical applicability. Future research will refine its mathematical properties, explore its performance on different tree types, and integrate it with existing bioinformatics tools for large-scale evolutionary analyses. The open source code is available at: https://pypi.org/project/parle/.

MOTIVATION: Noncoding RNAs (ncRNAs) express their functions by adopting molecular structures. Specifically, RNA secondary structures serve as a relatively stable intermediate step before tertiary structures, offering a reliable signature of molecular function. Consequently, within an RNA functional family, secondary structures are generally more evolutionarily conserved than sequences. Conversely, homologous RNA families grouped within an RNA clan share ancestors but typically exhibit structural differences. Inferring the evolution of RNA structures within RNA families and clans is crucial for gaining insights into functional adaptations over time and providing clues about the Ancient RNA World Hypothesis. RESULTS: We introduce the median problem and the small parsimony problem for ncRNA families, where secondary structures are represented as leaf-labeled trees. We utilize the Robinson-Foulds (RF) tree distance, which corresponds to a specific edit distance between RNA trees, and a new metric called the Internal-Leafset (IL) distance. While the RF tree distance compares sets of leaves descending from internal nodes of two RNA trees, the IL distance compares the collection of leaf-children of internal nodes. The latter is better at capturing differences in structural elements of RNAs than the RF distance, which is more focused on base pairs. We also consider a more general tree edit distance that allows the mapping of base pairs that are not perfectly aligned. We study the theoretical complexity of the median problem and the small parsimony problem under the three distance metrics and various biologically relevant constraints, and we present polynomial-time maximum parsimony algorithms for solving some versions of the problems. Our algorithms are applied to ncRNA families from the RFAM database, illustrating their practical utility. AVAILABILITY AND IMPLEMENTATION: https://github.com/bmarchand/rna\_small\_parsimony.

Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as Clique, Chromatic Number, and Steiner Tree has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.

For a graph class 𝒢, we define the 𝒢-modular cardinality of a graph G as the minimum size of a vertex partition of G into modules that each induces a graph in 𝒢. This generalizes other module-based graph parameters such as neighborhood diversity and iterated type partition. Moreover, if 𝒢 has bounded modular-width, the W[1]-hardness of a problem in 𝒢-modular cardinality implies hardness on modular-width, clique-width, and other related parameters. Several FPT algorithms based on modular partitions compute a solution table in each module, then combine each table into a global solution. This works well when each table has a succinct representation, but as we argue, when no such representation exists, the problem is typically W[1]-hard. We illustrate these ideas on the generic (α, β)-domination problem, which is a generalization of known domination problems such as Bounded Degree Deletion, k-Domination, and α-Domination. We show that for graph classes 𝒢 that require arbitrarily large solution tables, these problems are W[1]-hard in the 𝒢-modular cardinality, whereas they are fixed-parameter tractable when they admit succinct solution tables. This leads to several new positive and negative results for many domination problems parameterized by known and novel structural graph parameters such as clique-width, modular-width, and cluster-modular cardinality.

Abstract Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science, for example when the cohesive subgraph model considered is a clique. In this paper, we consider as a model of cohesive subgraph the 2-clubs, which are induced subgraphs of diameter at most 2. We prove new complexity results on the $$\mathsf {Min~2\text {-}Club~Cover}$$ Min2-ClubCover problem, a variant recently introduced in the literature which asks to cover the vertices of a graph with a minimum number of 2-clubs. First, we answer an open question on the decision version of $$\mathsf {Min~2\text {-}Club~Cover}$$ Min2-ClubCover that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W[1]-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of $$\mathsf {Min~2\text {-}Club~Cover}$$ Min2-ClubCover on some graph classes. We prove that $$\mathsf {Min~2\text {-}Club~Cover}$$ Min2-ClubCover remains NP-hard on subcubic planar graphs, W[2]-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution, and fixed-parameter tractable on graphs having bounded treewidth.

A graph G is a k -leaf power if there exists a tree T whose leaf set is V ( G ), and such that uv ∈ E ( G ) if and only if the distance between u and v in T is at most k (and u ≠ v ). The graph classes of k -leaf powers have several applications in computational biology, but recognizing them has remained a challenging algorithmic problem for the past two decades. The best known result is that 6-leaf powers can be recognized in polynomial time. In this article, we present an algorithm that decides whether a graph G is a k -leaf power in time O ( n f(k) for some function f that depends only on k (but has the growth rate of a power tower function). Our techniques are based on the fact that either a k -leaf power has a corresponding tree of low maximum degree, in which case finding it is easy, or every corresponding tree has large maximum degree. In the latter case, large-degree vertices in the tree imply that G has redundant substructures which can be pruned from the graph. In addition to solving a long-standing open problem, it is our hope that the structural results presented in this work can lead to further results on k -leaf powers and related classes.

In phylogenetic networks, picking a cherry consists of removing a leaf that shares a parent with another leaf, or removing a reticulate edge whose endpoints are parents of leaves. Cherry-picking operations were recently shown to have several structural and algorithmic applications in the study of networks, for instance in determining their reconstructibility or in solving the network hybridization and network containment problems. In particular, some networks within certain classes are isomorphic if they can be reduced to a single node by the same sequence of cherry-picking operations. Therefore, cherry-picking sequences contain information on the level of similarity between two networks. In this paper, we expand on this idea by devising four novel distances on networks based on cherry picking and their reverse operation. We provide bounds between these distances and show that three of them are equal despite their different formulations. We also show that computing these three equivalent distances is NP-hard, even when restricted to comparing a tree and a network. On the positive side, we show that they can be computed in quadratic time on two trees, providing a new comparative measure for phylogenetic trees that can be computed efficiently.

A graph G is a k-leaf power if there exists a tree T whose leaf set is V (G), and such that uv ∊ E(G) if and only if the distance between u and v in T is at most k. The graph classes of k-leaf powers have several applications in computational biology, but recognizing them has remained a challenging algorithmic problem for the past two decades. The best known result is that 6-leaf powers can be recognized in polynomial time. In this paper, we present an algorithm that decides whether a graph G is a k-leaf power in time O(nf(k)) for some function f that depends only on k (but has the growth rate of a power tower function). Our techniques are based on the fact that either a k-leaf power has a corresponding tree of low maximum degree, in which case finding it is easy, or every corresponding tree has large maximum degree. In the latter case, large degree vertices in the tree imply that G has redundant substructures which can be pruned from the graph. In addition to solving a longstanding open problem, we hope that the structural results presented in this work can lead to further results on k-leaf powers.

Horizontal gene transfer inference approaches are usually based on gene sequences: parametric methods search for patterns that deviate from a particular genomic signature, while phylogenetic methods use sequences to reconstruct the gene and species trees. However, it is well-known that sequences have difficulty identifying ancient transfers since mutations have enough time to erase all evidence of such events. In this work, we ask whether character-based methods can predict gene transfers. Their advantage over sequences is that homologous genes can have low DNA similarity, but still have retained enough important common motifs that allow them to have common character traits, for instance the same functional or expression profile. A phylogeny that has two separate clades that acquired the same character independently might indicate the presence of a transfer even in the absence of sequence similarity. We introduce perfect transfer networks, which are phylogenetic networks that can explain the character diversity of a set of taxa under the assumption that characters have unique births, and that once a character is gained it is rarely lost. Examples of such traits include transposable elements, biochemical markers and emergence of organelles, just to name a few. We study the differences between our model and two similar models: perfect phylogenetic networks and ancestral recombination networks. Our goals are to initiate a study on the structural and algorithmic properties of perfect transfer networks. We then show that in polynomial time, one can decide whether a given network is a valid explanation for a set of taxa, and show how, for a given tree, one can add transfer edges to it so that it explains a set of taxa. We finally provide lower and upper bounds on the number of transfers required to explain a set of taxa, in the worst case.

In computational biology, tandem duplication is an important biological phenomenon which can occur either at the genome or at the DNA level. A tandem duplication takes a copy of a genome segment and inserts it right after the segment---this can be represented as the string operation $AXB \Rightarrow AXXB$. Tandem exon duplications have been found in many species such as human, fly, and worm and have been largely studied in computational biology. The tandem duplication (TD) distance problem we investigate in this paper is defined as follows: given two strings $S$ and $T$ over the same alphabet $\Sigma$, compute the smallest sequence of TDs required to convert $S$ to $T$. The natural question of whether the TD distance can be computed in polynomial time was posed in 2004 by Leupold et al. and had remained open, despite the fact that TDs have received much attention ever since. In this paper, we focus on the special case when all characters of $S$ are distinct. This is known as the exemplar TD distance, which is of special relevance in bioinformatics. We first prove that this problem is NP-hard when the alphabet size is unbounded, settling the 16-year-old open problem. We then show how to adapt the proof to $|\Sigma|=4$, hence proving the NP-hardness of the TD problem for any $|\Sigma|\geq 4$. One of the tools we develop for the reduction is a new problem called Cost-Effective Subgraph, for which we obtain W[1]-hardness results that might be of independent interest. We finally show that computing the exemplar TD distance between $S$ and $T$ is fixed-parameter tractable. Our results open the door to many other questions, and we conclude with several open problems.

MOTIVATION: It is largely established that all extant mitochondria originated from a unique endosymbiotic event integrating an α-proteobacterial genome into an eukaryotic cell. Subsequently, eukaryote evolution has been marked by episodes of gene transfer, mainly from the mitochondria to the nucleus, resulting in a significant reduction of the mitochondrial genome, eventually completely disappearing in some lineages. However, in other lineages such as in land plants, a high variability in gene repertoire distribution, including genes encoded in both the nuclear and mitochondrial genome, is an indication of an ongoing process of Endosymbiotic Gene Transfer (EGT). Understanding how both nuclear and mitochondrial genomes have been shaped by gene loss, duplication and transfer is expected to shed light on a number of open questions regarding the evolution of eukaryotes, including rooting of the eukaryotic tree. RESULTS: We address the problem of inferring the evolution of a gene family through duplication, loss and EGT events, the latter considered as a special case of horizontal gene transfer occurring between the mitochondrial and nuclear genomes of the same species (in one direction or the other). We consider both EGT events resulting in maintaining (EGTcopy) or removing (EGTcut) the gene copy in the source genome. We present a linear-time algorithm for computing the DLE (Duplication, Loss and EGT) distance, as well as an optimal reconciled tree, for the unitary cost, and a dynamic programming algorithm allowing to output all optimal reconciliations for an arbitrary cost of operations. We illustrate the application of our EndoRex software and analyze different costs settings parameters on a plant dataset and discuss the resulting reconciled trees. AVAILABILITY AND IMPLEMENTATION: EndoRex implementation and supporting data are available on the GitHub repository via https://github.com/AEVO-lab/EndoRex.

Recently, due to the genomic sequence analysis in several types of cancer, genomic data based on copy number profiles (CNP for short) are getting more and more popular. A CNP is a vector where each component is a non-negative integer representing the number of copies of a specific segment of interest. The motivation is that in the late stage of certain types of cancer, the genomes are progressing rapidly by segmental duplications and deletions, and hence obtaining the exact sequences becomes difficult. Instead, the number of copies of important segments can be predicted from expression analysis and carries important biological information. Therefore, significant research has recently been devoted to the analysis of genomic data represented as CNP’s. In this paper, we present two streams of results. The first is the negative results on two open problems regarding the computational complexity of the Minimum Copy Number Generation (MCNG) problem posed by Qingge et al. in 2018. The Minimum Copy Number Generation (MCNG) is defined as follows: given a string S in which each character represents a gene or segment, and a CNP C, compute a string T from S, with the minimum number of segmental duplications and deletions, such that cnp(T)=C. It was shown by Qingge et al. that the problem is NP-hard if the duplications are tandem and they left the open question of whether the problem remains NP-hard if arbitrary duplications and/or deletions are used. We answer this question affirmatively in this paper; in fact, we prove that it is NP-hard to even obtain a constant factor approximation. This is achieved through a general-purpose lemma on set-cover reductions that require an exact cover in one direction, but not the other, which might be of independent interest. We also prove that the corresponding parameterized version is W[1]-hard, answering another open question by Qingge et al. The other result is positive and is based on a new (and more general) problem regarding CNP’s. The Copy Number Profile Conforming (CNPC) problem is formally defined as follows: given two CNP’s C₁ and C₂, compute two strings S₁ and S₂ with cnp(S₁)=C₁ and cnp(S₂)=C₂ such that the distance between S₁ and S₂, d(S₁,S₂), is minimized. Here, d(S₁,S₂) is a very general term, which means it could be any genome rearrangement distance (like reversal, transposition, and tandem duplication, etc). We make the first step by showing that if d(S₁,S₂) is measured by the breakpoint distance then the problem is polynomially solvable. We expect that this will trigger some related research along the line in the near future.

In computational biology, tandem duplication is an important biological phenomenon which can occur either at the genome or at the DNA level. A tandem duplication takes a copy of a genome segment and inserts it right after the segment - this can be represented as the string operation AXB ⇒ AXXB. Tandem exon duplications have been found in many species such as human, fly or worm, and have been largely studied in computational biology. The Tandem Duplication (TD) distance problem we investigate in this paper is defined as follows: given two strings S and T over the same alphabet, compute the smallest sequence of tandem duplications required to convert S to T. The natural question of whether the TD distance can be computed in polynomial time was posed in 2004 by Leupold et al. and had remained open, despite the fact that tandem duplications have received much attention ever since. In this paper, we prove that this problem is NP-hard, settling the 16-year old open problem. We further show that this hardness holds even if all characters of S are distinct. This is known as the exemplar TD distance, which is of special relevance in bioinformatics. One of the tools we develop for the reduction is a new problem called the Cost-Effective Subgraph, for which we obtain W[1]-hardness results that might be of independent interest. We finally show that computing the exemplar TD distance between S and T is fixed-parameter tractable. Our results open the door to many other questions, and we conclude with several open problems.

Consider two robots that start at the origin of the infinite line in search of an exit at an unknown location on the line. The robots can only communicate if they arrive at the same location at exactly the same time, i.e. they use the so-called face-to-face communication model. The group search time is defined as the worst-case time as a function of $d$, the distance of the exit from the origin, when both robots can reach the exit. It has long been known that for a single robot traveling at unit speed, the search time is at least $9d-o(d)$. It was shown recently that $k\geq2$ robots traveling at unit speed also require at least $9d$ group search time. We investigate energy-time trade-offs in group search by two robots, where the energy loss experienced by a robot traveling a distance $x$ at constant speed $s$ is given by $s^2 x$. Specifically, we consider the problem of minimizing the total energy used by the robots, under the constraints that the search time is at most a multiple $c$ of the distance $d$ and the speed of the robots is bounded by $b$. Motivation for this study is that for the case when robots must complete the search in $9d$ time with maximum speed one, a single robot requires at least $9d$ energy, while for two robots, all previously proposed algorithms consume at least $28d/3$ energy. When the robots have bounded memory, we generalize existing algorithms to obtain a family of optimal (and in some cases nearly optimal) algorithms parametrized by pairs of $b,c$ values that can solve the problem for the entire spectrum of these pairs for which the problem is solvable. We also propose a novel search algorithm, with unbounded memory, that simultaneously achieves search time $9d$ and consumes energy $8.42588d$. Our result shows that two robots can search on the line in optimal time $9d$ while consuming less total energy than a single robot within the same search time.

Reconstructing ancestral gene orders in a given phylogeny is a classical problem in comparative genomics. Most existing methods compare conserved features in extant genomes in the phylogeny to define potential ancestral gene adjacencies, and either try to reconstruct all ancestral genomes under a global evolutionary parsimony criterion, or, focusing on a single ancestral genome, use a scaffolding approach to select a subset of ancestral gene adjacencies, generally aiming at reducing the fragmentation of the reconstructed ancestral genome. In this paper, we describe an exact algorithm for the Small Parsimony Problem that combines both approaches. We consider that gene adjacencies at internal nodes of the species phylogeny are weighted, and we introduce an objective function defined as a convex combination of these weights and the evolutionary cost under the Single-Cut-or-Join (SCJ) model. The weights of ancestral gene adjacencies can, e.g., be obtained through the recent availability of ancient DNA sequencing data, which provide a direct hint at the genome structure of the considered ancestor, or through probabilistic analysis of gene adjacencies evolution. We show the NP-hardness of our problem variant and propose a Fixed-Parameter Tractable algorithm based on the Sankoff-Rousseau dynamic programming algorithm that also allows to sample co-optimal solutions. We apply our approach to mammalian and bacterial data providing different degrees of complexity. We show that including adjacency weights in the objective has a significant impact in reducing the fragmentation of the reconstructed ancestral gene orders. An implementation is available at http://github.com/nluhmann/PhySca.

Reconciling gene trees with a species tree is a fundamental problem to understand the evolution of gene families. Many existing approaches reconcile each gene tree independently. However, it is well-known that the evolution of gene families is interconnected. In this paper, we extend a previous approach to reconcile a set of gene trees with a species tree based on segmental macro-evolutionary events, where segmental duplication events and losses are associated with cost delta and lambda, respectively. We show that the problem is polynomial-time solvable when delta <= lambda (via LCA-mapping), while if delta > lambda the problem is NP-hard, even when lambda = 0 and a single gene tree is given, solving a long standing open problem on the complexity of the reconciliation problem. On the positive side, we give a fixed-parameter algorithm for the problem, where the parameters are delta/lambda and the number d of segmental duplications, of time complexity O(ceil[delta/lambda]^d * n * delta/lambda). Finally, we demonstrate the usefulness of this algorithm on two previously studied real datasets: we first show that our method can be used to confirm or refute hypothetical segmental duplications on a set of 16 eukaryotes, then show how we can detect whole genome duplications in yeast genomes.

Motivation: When gene duplication occurs, one of the copies may become free of selective pressure and evolve at an accelerated pace. This has important consequences on the prediction of orthology relationships, since two orthologous genes separated by divergence after duplication may differ in both sequence and function. In this work, we make the distinction between the primary orthologs, which have not been affected by accelerated mutation rates on their evolutionary path, and the secondary orthologs, which have. Similarity-based prediction methods will tend to miss secondary orthologs, whereas phylogeny-based methods cannot separate primary and secondary orthologs. However, both types of orthology have applications in important areas such as gene function prediction and phylogenetic reconstruction, motivating the need for methods that can distinguish the two types. Results: We formalize the notion of divergence after duplication and provide a theoretical basis for the inference of primary and secondary orthologs. We then put these ideas to practice with the Hybrid Prediction of Paralogs and Orthologs (HyPPO) framework, which combines ideas from both similarity and phylogeny approaches. We apply our method to simulated and empirical datasets and show that we achieve superior accuracy in predicting primary orthologs, secondary orthologs and paralogs. Availability and implementation: HyPPO is a modular framework with a core developed in Python and is provided with a variety of C++ modules. The source code is available at https://github.com/manuellafond/HyPPO. Supplementary information: Supplementary data are available at Bioinformatics online.

The supertree problem asking for a tree displaying a set of consistent input trees has been largely considered for the reconstruction of species trees. Here, we rather explore this framework for the sake of reconstructing a gene tree from a set of input gene trees on partial data. In this perspective, the phylogenetic tree for the species containing the genes of interest can be used to choose among the many possible compatible "supergenetrees", the most natural criteria being to minimize a reconciliation cost. We develop a variety of algorithmic solutions for the construction and correction of gene trees using the supertree framework. A dynamic programming supertree algorithm for constructing or correcting gene trees, exponential in the number of input trees, is first developed for the less constrained version of the problem. It is then adapted to gene trees with nodes labeled as duplication or speciation, the additional constraint being to preserve the orthology and paralogy relations between genes. Then, a quadratic time algorithm is developed for efficiently correcting an initial gene tree while preserving a set of "trusted" subtrees, as well as the relative phylogenetic distance between them, in both cases of labeled or unlabeled input trees. By applying these algorithms to the set of Ensembl gene trees, we show that this new correction framework is particularly useful to correct weakly-supported duplication nodes. The C++ source code for the algorithms and simulations described in the paper are available at https://github.com/UdeM-LBIT/SuGeT.

Determining the evolutionary history of a given biological data is an important task in biological sciences. Given a set of quartet topologies over a set of taxa, the Maximum Quartet Consistency (MQC) problem consists of computing a global phylogeny that satisfies the maximum number of quartets. A number of solutions have been proposed for the MQC problem, including Dynamic Programming, Constraint Programming, and more recently Answer Set Programming (ASP). ASP is currently the most efficient approach for optimally solving the MQC problem. This paper proposes encoding the MQC problem with pseudo-Boolean (PB) constraints. The use of PB allows solving the MQC problem with efficient PB solvers, and also allows considering different modeling approaches for the MQC problem. Initial results are promising, and suggest that PB can be an effective alternative for solving the MQC problem.

We consider a problem of routing on directed paths and trees to a single destination, with rate-limited, adversarial traffic. In particular, we focus on local buffer management algorithms that ensure no packet loss, while minimizing the size of the required buffers. While a centralized algorithm for the problem that uses constant-sized buffers has been recently shown [21], there is no known local algorithm that achieves a sub-linear buffer size. In this paper we show tight bounds for the maximum buffer size needed by l-local algorithms for information gathering on directed paths and trees, where an algorithm is called l-local if the decision made by each node v depends only on the sizes of the buffers at most l hops away from v.

Phylogenetic tree reconstruction is usually done by local search heuristics that explore the space of the possible tree topologies via simple rearrangements of their structure. Tree rearrangement heuristics have been used in combination with practically all optimization criteria in use, from maximum likelihood and parsimony to distance-based principles, and in a Bayesian context. Their basic components are rearrangement moves that specify all possible ways of generating alternative phylogenies from a given one, and whose fundamental property is to be able to transform, by repeated application, any phylogeny into any other phylogeny. Despite their long tradition in tree-based phylogenetics, very little research has gone into studying similar rearrangement operations for phylogenetic network-that is, phylogenies explicitly representing scenarios that include reticulate events such as hybridization, horizontal gene transfer, population admixture, and recombination. To fill this gap, we propose "horizontal" moves that ensure that every network of a certain complexity can be reached from any other network of the same complexity, and "vertical" moves that ensure reachability between networks of different complexities. When applied to phylogenetic trees, our horizontal moves-named rNNI and rSPR-reduce to the best-known moves on rooted phylogenetic trees, nearest-neighbor interchange and rooted subtree pruning and regrafting. Besides a number of reachability results-separating the contributions of horizontal and vertical moves-we prove that rNNI moves are local versions of rSPR moves, and provide bounds on the sizes of the rNNI neighborhoods. The paper focuses on the most biologically meaningful versions of phylogenetic networks, where edges are oriented and reticulation events clearly identified. Moreover, our rearrangement moves are robust to the fact that networks with higher complexity usually allow a better fit with the data. Our goal is to provide a solid basis for practical phylogenetic network reconstruction.

Polytomies in gene trees are multifurcated nodes corresponding to unresolved parts of the tree, usually due to insufficient differentiation between sequences of homologous gene copies. Apart from gene sequences, other information such as that contained in the species tree can be used to resolve such intricate parts of a gene tree. The problem of resolving a multifurcated tree has been considered by many authors, the objective function often being the number of duplications and losses reflected by the reconciliation of the resolved gene tree with the species tree. Here, we present PolytomySolver, an algorithm accounting for a more general model allowing different costs for duplications and losses per species. The time complexity of this algorithm is linear for the unit cost and is quadratic for the general cost, which outperforms the best known solutions so far by a linear factor. We show on simulated trees that the gain in theoretical complexity has a real practical impact on running times.

Combining a set of trees on partial datasets into a single tree is a classical method for inferring large phylogenetic trees. Ideally, the combined tree should display each input partial tree, which is only possible if input trees do not contain contradictory phylogenetic information. The simplest version of the supertree problem is thus to state whether a set of trees is compatible, and if so, construct a tree displaying them all. Classically, supertree methods have been applied to the reconstruction of species trees. Here we rather consider reconstructing a super gene tree in light of a known species tree S. We define the supergenetree problem as finding, among all supertrees displaying a set of input gene trees, one supertree minimizing a reconciliation distance with S. We first show how classical exact methods to the supertree problem can be extended to the supergenetree problem. As all these methods are highly exponential, we also exhibit a natural greedy heuristic for the duplication cost, based on minimizing the set of duplications preceding the first speciation event. We then show that both the supergenetree problem and its restriction to minimizing duplications preceding the first speciation are NP-hard to approximate within a n1-ϵ factor, for any 0 < ϵ < 1. Finally, we show that a restriction of this problem to uniquely labeled speciation gene trees, which is relevant to many biological applications, is also NP-hard. Therefore, we introduce new avenues in the field of supertrees, and set the theoretical basis for the exploration of various algorithmic aspects of the problems.

In 1990, Hendry Conjectured that every Hamiltonian chordal graph is cycle extendable; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on $n$ vertices for any $n \geq 15$. Furthermore, we show that there exist counterexamples where the ratio of the length of a nonextendable cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendability in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably $P_n$ and the bull.

MOTIVATION: Large-scale methods for inferring gene trees are error-prone. Correcting gene trees for weakly supported features often results in non-binary trees, i.e. trees with polytomies, thus raising the natural question of refining such polytomies into binary trees. A feature pointing toward potential errors in gene trees are duplications that are not supported by the presence of multiple gene copies. RESULTS: We introduce the problem of refining polytomies in a gene tree while minimizing the number of created non-apparent duplications in the resulting tree. We show that this problem can be described as a graph-theoretical optimization problem. We provide a bounded heuristic with guaranteed optimality for well-characterized instances. We apply our algorithm to a set of ray-finned fish gene trees from the Ensembl database to illustrate its ability to correct dubious duplications. AVAILABILITY AND IMPLEMENTATION: The C++ source code for the algorithms and simulations described in the article are available at http://www-ens.iro.umontreal.ca/~lafonman/software.php. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

In today's age of exposure, websites and Internet services are collecting personal data-with or without the knowledge or consent of users. Not only does new technology provide an abundance of methods for organizations to gather and store information, but people are also willingly sharing data with increasing frequency, exposing their intimate lives on social media websites such as Facebook, Twitter, You tube, My space and others. Moreover, online data brokers, search engines, data aggregators and many other actors of the web are profiling people for various purposes such as the improvement of marketing through better statistics and an ability to predict consumer behaviour. Other less known reasons include understanding the newest trends in education, gathering people's medical history or observing tendencies in political opinions. People who care about privacy use the Privacy Enhancing Technologies (PETs) to protect their data, even though clearly not sufficiently. Indeed, as soon as information is recorded in a database, it becomes permanently available for analysis. Consequently even the most privacy aware users are not safe from the threat of re-identification. On the other hand, there are many people who are willing to share their personal information, even when fully conscious of the consequences. A claim from the advocates of open access information is that the preservation of privacy should not be an issue, as people seem to be confortable in a world where their tastes, lifestyle or personality are digitized and publicly available. This paper deals with Internet data collection and voluntary information disclosure, with an emphasis on the problems and challenges facing privacy nowadays.

BACKGROUND: Reconciled gene trees yield orthology and paralogy relationships between genes. This information may however contradict other information on orthology and paralogy provided by other footprints of evolution, such as conserved synteny. RESULTS: We explore a way to include external information on orthology in the process of gene tree construction. Given an initial gene tree and a set of orthology constraints on pairs of genes or on clades, we give polynomial-time algorithms for producing a modified gene tree satisfying the set of constraints, that is as close as possible to the original one according to the Robinson-Foulds distance. We assess the validity of the modifications we propose by computing the likelihood ratio between initial and modified trees according to sequence alignments on Ensembl trees, showing that often the two trees are statistically equivalent. AVAILABILITY: Software and data available upon request to the corresponding author.

It is not uncommon for business managers to use recent innovations in information and communications technology to monitor employees and job candidates. These methods not only rely on heavy surveillance during working hours of employees but can also be applied outside their professional environment, to impinge on their personal lives. Surveillance techniques encompass such traditional means like recording cameras to more recent methods including analyzing social networks pages, performing extensive web searches and dealing with online data brokers. While monitoring initiatives set up by employers can have benefits for companies, the threat to privacy they entail can deteriorate the mental and physical health of employees and have a negative impact on the quality of relationship between colleagues. Businesses have a social responsibility and need to ensure that their behavior does not infringe upon their employee's rights to privacy. In this non-technical paper, we discuss some online approaches adopted by companies regarding employee surveillance. We elaborate on various methods employed by managers to monitor their employees and gain as much information as possible on job candidates. Then, these techniques are further discussed from the standpoint of their moral and legal perspectives with regards to privacy rights.