• In this paper we study a variant of Vertex Cover where the activities of vertices are characterized by time intervals. We explore a scenario where the temporal span of each vertex’s activity interval is bounded by an integer, and the objective is to maximize the number of (temporal) edges that are covered. • We establish the APX-hardness of this problem and the NP-hardness of the corresponding decision problem, even under the restricted conditions where: the temporal domain comprises only two timestamps and each edge appears at most once and; no two edges are associated to a same label. • We delve into the parameterized complexity of the problem, offering two fixed-parameter algorithms parameterized by: the number k of temporal edges covered by the solution, and the number h of temporal edges left uncovered by the solution. • We focus again on the approximability of the problem and present a polynomial-time approximation algorithm achieving a factor of 3 4 . Different variants of Vertex Cover have recently garnered attention in the context of temporal graphs. One of these variants is motivated by the need to summarize timeline activities in social networks. Here, the activities of individual vertices, representing users, are characterized by time intervals. In this paper, we explore a scenario where the temporal span of each vertex’s activity interval is bounded by an integer ℓ, and the objective is to maximize the number of (temporal) edges that are covered. We establish the APX-hardness of this problem and the NP-hardness of the corresponding decision problem, even under the restricted conditions where: the temporal domain comprises only two timestamps and each edge appears at most once and; no two edges are associated to a same label. Subsequently, we delve into the parameterized complexity of the problem, offering two fixed-parameter algorithms parameterized by: (i) the number k of temporal edges covered by the solution, and (ii) the number h of temporal edges not covered by the solution. Finally, we present a polynomial-time approximation algorithm achieving a factor of 3 4 .

In this contribution we introduce two combinatorial problems related to graph string matching, motivated by recent approaches in computational genomics. Given a DAG where each node is labeled by a symbol, the problems aim to find a path in the DAG whose nodes contain all (or the maximum number of) symbols of the alphabet. We introduce a decision problem, Σ-Representing Path , that asks whether there exists a path that contains all the symbols of the alphabet, and an optimization problem, called Maximum Representing Path , that asks for a path that contains the maximum number of symbols. We analyze the complexity of the problems, showing the NP-completeness of Σ-Representing Path when each symbol labels at most three nodes in the DAG, and showing the APX-hardness of Maximum Representing Path when each symbol labels at most two nodes in the DAG. We complement the first result by giving a polynomial-time algorithm for Σ-Representing Path when each symbol labels at most two nodes in the DAG. Then we investigate the parameterized complexity of the two problems for two parameters: (1) the number of symbols in a solution and (2) the distance from a set of disjoint paths. We show that both problems are FPT when parameterized by the former parameter, and W[1]-hard for the latter. We consider the approximation of Maximum Representing Path , and we give an approximation algorithm of factor the maximum number of occurrences of a symbol and an approximation algorithm of factor O P T , where OPT is the number of distinct symbols in an optimal solution. We also show that Maximum Representing Path cannot be approximated within factor e e − 1 − α , for any constant α > 0, unless NP ⊆ DTIME (| V | O (log log | V |) ) ( V is the set of nodes of the DAG).

We consider a variant of the Vertex Cover problem on temporal graphs, called Minimum Timeline Cover (k-MinTimelineCover). Temporal graphs are used to model complex systems, describing how edges (relations) change in a discrete time domain. The k-MinTimelineCover problem has been introduced in complex data summarization and synthesis jobs. Given a temporal graph G, k-MinTimelineCover asks to define k activity intervals for each vertex, such that each temporal edge is covered by at least one active interval. The objective function is the minimization of the sum of interval lengths. k-MinTimelineCover is NP-hard and even hard to approximate within any factor for k > 1. While the literature has mainly focused on the cases k = 1, in this contribution we consider the case k > 1. We first present an ILP formulation that is able to solve the problem on moderate size instances. Then we develop an efficient heuristic, based on local search which is built on top of the solution of an existing literature method. Finally, we present an experimental evaluation of our algorithms on synthetic data sets, that shows in particular that our heuristic has a consistent improvement on the state-of-the art method.

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.

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.

In this contribution we consider a variant of the vertex cover problem in temporal graphs that has been recently introduced to summarize timeline activities in social networks. The problem is NP-hard, even when the time domain considered consists of two timestamps. We further analyze the complexity of this problem, focusing on temporal graphs of bounded degree. We prove that the problem is NP-hard when (1) each vertex has degree at most one in each timestamp and (2) each vertex is connected with at most three neighbors, has degree at most two in each timestamp and the time domain consists of three timestamps. On the other hand, we prove that the problem is in P when each vertex is connected with at most two neighbors. Then we present a fixed-parameter algorithm for the restriction where we bound the number of interactions in each timestamp and the length of the interval where a vertex has incident temporal edges.

Longest Run Subsequence is a problem introduced recently in the context of the scaffolding phase of genome assembly (Schrinner et al., WABI 2020). The problem asks for a maximum length subsequence of a given string that contains at most one run for each symbol (a run is a maximum substring of consecutive identical symbols). The problem has been shown to be NP-hard and to be fixed-parameter tractable when the parameter is the size of the alphabet on which the input string is defined. In this paper we further investigate the complexity of the problem and we show that it is fixed-parameter tractable when it is parameterized by the number of runs in a solution, a smaller parameter. Moreover, we investigate the kernelization complexity of Longest Run Subsequence and we prove that it does not admit a polynomial kernel when parameterized by the size of the alphabet or by the number of runs. Finally, we consider the restriction of Longest Run Subsequence when each symbol has at most two occurrences in the input string and we show that it is APX-hard.

The evolution of networks is a fundamental topic in network analysis and mining. One of the approaches that has been recently considered in this field is the analysis of temporal networks, where relations between elements can change over time. A relevant problem in the analysis of temporal networks is the identification of cohesive or dense subgraphs since they are related to communities. In this contribution, we present a method based on genetic algorithms and on a greedy heuristic to identify dense subgraphs in a temporal network. We present experimental results considering both synthetic and real-networks, and we analyze the performance of the proposed method when varying the size of the population and the number of generations. The experimental results show that our heuristic generally performs better in terms of quality of the solutions than the state-of-art method for this problem. On the other hand, the state-of-art method is faster, although comparable with our method, when the size of the population and the number of generations are limited to small values.

Recommender Systems have became extremely appealing for all technology enhanced learning researches aimed to design, develop and test technical innovations which support and enhance learning and teaching practices of both individuals and organizations. In this scenario a new emerging paradigm of explainable Recommander Systems leverages social friend information to provide (social) explanations in order to supply users with his/her friends’ public interests as explained recommendation. In this paper we introduce our educational platform called “WhoTeach”, an innovative and original system to integrate knowledge discovery, social networks analysis, and educational services. In particular, we report here our work in progress for providing “WhoTeach” environment with optimized Social Explainable Recommandations oriented to design new teachers’ programmes and courses.

Social interaction allows to support the disease management by creating online spaces where patients can interact with clinicians, and share experiences with other patients. Therefore, promoting targeted communication in online social spaces is a means to group patients around shared goals, offer emotional support, and finally engage patients in their healthcare decision making process. In this paper, we approach the argument from a theoretical perspective: we design an optimization problem aimed to encourage the creation of (induced) sub-networks of patients which, being recently diagnosed, wish to deepen the knowledge about their medical treatment with some other similar profiled patients, which have already been followed up by specific (even alternative) care centers. In particular, due to the computational hardness of the proposed problem, we provide approximated solutions based on distributed heuristics (i.e., Genetic Algorithms). Results are given for simulated data using Erdos-Renyi random graphs.

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.

Inspired by a recent approach for genome reconstruction from incomplete data, we consider a variant of the longest common subsequence problem for the comparison of two sequences, one of which is incomplete, i.e. it has some missing elements. The new combinatorial problem, called Longest Filled Common Subsequence, given two sequences A and B, and a multiset M of symbols missing in B, asks for a sequence B* obtained by inserting the symbols of M into B so that B* induces a common subsequence with A of maximum length. First, we investigate the computational and approximation complexity of the problem and we show that it is NP-hard and APX-hard when A contains at most two occurrences of each symbol. Then, we give a 3/5 approximation algorithm for the problem. Finally, we present a fixed-parameter algorithm, when the problem is parameterized by the number of symbols inserted in B that "match" symbols of A.

MOTIVATION: Haplotype assembly is the computational problem of reconstructing haplotypes in diploid organisms and is of fundamental importance for characterizing the effects of single-nucleotide polymorphisms on the expression of phenotypic traits. Haplotype assembly highly benefits from the advent of 'future-generation' sequencing technologies and their capability to produce long reads at increasing coverage. Existing methods are not able to deal with such data in a fully satisfactory way, either because accuracy or performances degrade as read length and sequencing coverage increase or because they are based on restrictive assumptions. RESULTS: By exploiting a feature of future-generation technologies-the uniform distribution of sequencing errors-we designed an exact algorithm, called HapCol, that is exponential in the maximum number of corrections for each single-nucleotide polymorphism position and that minimizes the overall error-correction score. We performed an experimental analysis, comparing HapCol with the current state-of-the-art combinatorial methods both on real and simulated data. On a standard benchmark of real data, we show that HapCol is competitive with state-of-the-art methods, improving the accuracy and the number of phased positions. Furthermore, experiments on realistically simulated datasets revealed that HapCol requires significantly less computing resources, especially memory. Thanks to its computational efficiency, HapCol can overcome the limits of previous approaches, allowing to phase datasets with higher coverage and without the traditional all-heterozygous assumption. AVAILABILITY AND IMPLEMENTATION: Our source code is available under the terms of the GNU General Public License at http://hapcol.algolab.eu/ CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

The Minimum Path Cover (MinPC) problem on directed acyclic graphs (DAGs) is a classical problem in graph theory that provides a clear and simple mathematical formulation for several applications in computational biology. In this paper, we study the computational complexity of three constrained variants of MinPC motivated by the recent introduction of Next-Generation Sequencing technologies. The first variant (MinRPC), given a DAG and a set of pairs of vertices, asks for a minimum-cardinality set of (not necessarily disjoint) paths such that both vertices of each pair belong to the same path. For this problem, we establish a sharp tractability borderline depending on the ‘overlapping degree’ of the instance, a natural parameter in some applications of the problem. The second variant we consider (MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum-cardinality set of (not necessarily disjoint) paths ‘covering’ all the vertices of the graph and such that both vertices of each pair belong to the same path. For this problem, we show that, while it is NP-hard to compute if there exists a solution consisting of at most three paths, it is possible to decide in polynomial time whether a solution consisting of at most two paths exists. The third variant (MaxRPSP), given a DAG and a set of pairs of vertices, asks for a single path containing the maximum number of the given pairs of vertices. We show that MaxRPSP is W[1]-hard when parameterized by the number of covered pairs and we give a fixed-parameter algorithm when the parameter is the maximum overlapping degree.

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.

The haplotype resolution from xor-genotype data has been recently formulated as a new model for genetic studies. The xor-genotype data is a cheaply obtainable type of data distinguishing heterozygous from homozygous sites without identifying the homozygous alleles. In this paper, we propose a formulation based on a well-known model used in haplotype inference: pure parsimony. We exhibit exact solutions of the problem by providing polynomial time algorithms for some restricted cases and a fixed-parameter algorithm for the general case. These results are based on some interesting combinatorial properties of a graph representation of the solutions. Furthermore, we show that the problem has a polynomial time k-approximation, where k is the maximum number of xor-genotypes containing a given single nucleotide polymorphisms (SNP). Finally, we propose a heuristic and produce an experimental analysis showing that it scales to real-world large instances taken from the HapMap project.

In this paper, we investigate the computational and approximation complexity of the Exemplar Longest Common Subsequence of a set of sequences (ELCS problem), a generalization of the Longest Common Subsequence problem, where the input sequences are over the union of two disjoint sets of symbols, a set of mandatory symbols and a set of optional symbols. We show that different versions of the problem are APX-hard even for instances with two sequences. Moreover, we show that the related problem of determining the existence of a feasible solution of the Exemplar Longest Common Subsequence of two sequences is NP-hard. On the positive side, we first present an efficient algorithm for the ELCS problem over instances of two sequences where each mandatory symbol can appear in total at most three times in the sequences. Furthermore, we present two fixed-parameter algorithms for the ELCS problem over instances of two sequences where the parameter is the number of mandatory symbols.

Most of the available knowledge management systems pay little attention to two important aspects: the need of supporting emerging communities of interest together with the official organizational structure; and the need of cluing together knowledge associated with any kind of involved entity including people, communities, and informal knowledge. The MILK system enhances knowledge discovery and sharing by providing services addressing these aspects and supplying innovative interfaces and interaction styles. The goal of MILK is to become a familiar environment integrated in the every-day activities of dynamic modern workers. To meet the users' needs, the solution proposed by MILK roots in ethnographic analysis capturing the common practices within an organization.