Computer Science

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{ó}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings. In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vu{\v c}kovi{ć} can be adapted to prove results on the product version. We prove that 4-chromatic graphs verify the product version of the 1-2-3 Conjecture. We also prove that for all graphs we can design 3-labellings that almost have the desired property. This leads to a new problem, that we solve for some graph classes.

We introduce a very natural generalization of the well-known problem of simultaneous congruences. Instead of searching for a positive integer s that is specified by n fixed remainders modulo integer divisors a 1 ,…, a n we consider remainder intervals R 1 ,…, R n such that s is feasible if and only if s is congruent to r i modulo a i for some remainder r i in interval R i for all i . This problem is a special case of a 2-stage integer program with only two variables per constraint which is is closely related to directed Diophantine approximation as well as the mixing set problem. We give a hardness result showing that the problem is NP-hard in general. By investigating the case of harmonic divisors, i.e. a i+1 / a i is an integer for all i<n , which was heavily studied for the mixing set problem as well, we also answer a recent algorithmic question from the field of real-time systems. We present an algorithm to decide the feasibility of an instance in time O( n 2 ) and we show that if it exists even the smallest feasible solution can be computed in strongly polynomial time O( n 3 ) .

Cops and Robbers games have been studied for the last few decades in computer science and mathematics. As in general pursuit evasion games, pursuers (cops) seek to capture evaders (robbers); however, players move in turn and are constrained to move on a discrete structure, usually a graph, and know the exact location of their opponent. In 2017, Bonato and MacGillivray presented a general characterization of Cops and Robbers games in order for them to be globally studied. However, their model doesn't cover cases where stochastic events may occur, such as the robbers moving in a random fashion. In this paper we present a novel model with stochastic elements that we call a Generalized Probabilistic Cops and Robbers game (GPCR). A typical such game is one where the robber moves according to a probabilistic distribution, either because she is rather lost or drunk than evading, or because she is a robot. We present results to solve GPCR games, thus enabling one to study properties relating to the optimal strategies in large classes of Cops and Robbers games. Some classic Cops and Robbers games properties are also extended.

Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps ε that assign a subset of colors to each pair of vertices in X and that can be explained by a tree T with edges that are labeled with subsets of colors in the sense that the color m appears in ε(x,y) if and only if m appears in a label along the unique path between x and y in T . We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where |ε(x,y)|≤1 the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.

In this paper, we study the generalized gapped k-mer filters and derive a closed form solution for their coefficients. We consider nonnegative integers ??and k , with k?��? , and an ??-tuple B=( b 1 ,?? b ??) of integers b i ?? , i=1,????. We introduce and study an incidence matrix A= A ??k;B . We develop a Möbius-like function ν B which helps us to obtain closed forms for a complete set of mutually orthogonal eigenvectors of A ??A as well as a complete set of mutually orthogonal eigenvectors of A A ??corresponding to nonzero eigenvalues. The reduced singular value decomposition of A and combinatorial interpretations for the nullity and rank of A , are among the consequences of this approach. We then combine the obtained formulas, some results from linear algebra, and combinatorial identities of elementary symmetric functions and ν B , to provide the entries of the Moore-Penrose pseudo-inverse matrix A + and the Gapped k-mer filter matrix A + A .

Given a CNF formula Φ with clauses C 1 ,…, C m and variables V={ x 1 ,…, x n } , a truth assignment a:V→{0,1} of Φ leads to a clause sequence σ Φ (a)=( C 1 (a),…, C m (a))∈{0,1 } m where C i (a)=1 if clause C i evaluates to 1 under assignment a , otherwise C i (a)=0 . The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties. We consider a problem posed at Dagstuhl Seminar 19211 "Enumeration in Data Management" (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.

The bigraph theory is a relatively young, yet formally rigorous, mathematical framework encompassing Robin Milner's previous work on process calculi, on the one hand, and provides a generic meta-model for complex systems such as multi-agent systems, on the other. A bigraph F=⟨ F P , F L ⟩ is a superposition of two independent graph structures comprising a place graph F P (i.e., a forest) and a link graph F L (i.e., a hypergraph), sharing the same node set, to express locality and communication of processes independently from each other. In this paper, we take some preparatory steps towards an algorithm for generating random bigraphs with preferential attachment feature w.r.t. F P and assortative (disassortative) linkage pattern w.r.t. F L . We employ parameters allowing one to fine-tune the characteristics of the generated bigraph structures. To study the pattern formation properties of our algorithmic model, we analyze several metrics from graph theory based on artificially created bigraphs under different configurations. Bigraphs provide a quite useful and expressive semantic for process calculi for mobile and global ubiquitous computing. So far, this subject has not received attention in the bigraph-related scientific literature. However, artificial models may be particularly useful for simulation and evaluation of real-world applications in ubiquitous systems necessitating random structures.

Genome assembly asks to reconstruct an unknown string from many shorter substrings of it. Even though it is one of the key problems in Bioinformatics, it is generally lacking major theoretical advances. Its hardness stems both from practical issues (size and errors of real data), and from the fact that problem formulations inherently admit multiple solutions. Given these, at their core, most state-of-the-art assemblers are based on finding non-branching paths (unitigs) in an assembly graph. If one defines a genome assembly solution as a closed arc-covering walk of the graph, then unitigs appear in all solutions, being thus safe partial solutions. All all such safe walks were recently characterized as omnitigs, leading to the first safe and complete genome assembly algorithm. Even if omnitig finding was improved to quadratic time, it remained open whether the crucial linear-time feature of finding unitigs can be attained with omnitigs. We describe a surprising O(m) -time algorithm to identify all maximal omnitigs of a graph with n nodes and m arcs, notwithstanding the existence of families of graphs with Θ(mn) total maximal omnitig size. This is based on the discovery of a family of walks (macrotigs) with the property that all the non-trivial omnitigs are univocal extensions of subwalks of a macrotig, with two consequences: (1) A linear-time output-sensitive algorithm enumerating all maximal omnitigs. (2) A compact O(m) representation of all maximal omnitigs, which allows, e.g., for O(m) -time computation of various statistics on them. Our results close a long-standing theoretical question inspired by practical genome assemblers, originating with the use of unitigs in 1995. We envision our results to be at the core of a reverse transfer from theory to practical and complete genome assembly programs, as has been the case for other key Bioinformatics problems.

With a view on graph clustering, we present a definition of vertex-to-vertex distance which is based on shared connectivity. We argue that vertices sharing more connections are closer to each other than vertices sharing fewer connections. Our thesis is centered on the widely accepted notion that strong clusters are formed by high levels of induced subgraph density, where subgraphs represent clusters. We argue these clusters are formed by grouping vertices deemed to be similar in their connectivity. At the cluster level (induced subgraph level), our thesis translates into low mean intra-cluster distances. Our definition differs from the usual shortest-path geodesic distance. In this article, we compare three distance measures from the literature. Our benchmark is the accuracy of each measure's reflection of intra-cluster density, when aggregated (averaged) at the cluster level. We conduct our tests on synthetic graphs generated using the planted partition model, where clusters and intra-cluster density are known in advance. We examine correlations between mean intra-cluster distances and intra-cluster densities. Our numerical experiments show that Jaccard and Otsuka-Ochiai offer very accurate measures of density, when averaged over vertex pairs within clusters.

We consider the problem of collective exploration of a known n -node edge-weighted graph by k mobile agents that have limited energy but are capable of energy transfers. The agents are initially placed at an arbitrary subset of nodes in the graph, and each agent has an initial, possibly different, amount of energy. The goal of the exploration problem is for every edge in the graph to be traversed by at least one agent. The amount of energy used by an agent to travel distance x is proportional to x . In our model, the agents can {\em share} energy when co-located: when two agents meet, one can transfer part of its energy to the other. For an n -node path, we give an O(n+k) time algorithm that either finds an exploration strategy, or reports that one does not exist. For an n -node tree with ??leaves, we give an O(n+??k 2 ) algorithm that finds an exploration strategy if one exists. Finally, for the general graph case, we show that the problem of deciding if exploration is possible by energy-sharing agents is NP-hard, even for 3-regular graphs. In addition, we show that it is always possible to find an exploration strategy if the total energy of the agents is at least twice the total weight of the edges; moreover, this is asymptotically optimal.