Computer Science

Functions correspond to one of the key concepts in mathematics and science, allowing the representation and modeling of several types of signals and systems. The present work develops an approach for characterizing the coverage and interrelationship between discrete signals that can be fitted by a set of reference functions, allowing the definition of transition networks between the considered discrete signals. While the adjacency between discrete signals is defined in terms of respective Euclidean distances, the property of being adjustable by the reference functions provides an additional constraint leading to a surprisingly diversity of transition networks topologies. First, we motivate the possibility to define transitions between parametric continuous functions, a concept that is subsequently extended to discrete functions and signals. Given that the set of all possible discrete signals in a bound region corresponds to a finite number of cases, it becomes feasible to verify the adherence of each of these signals with respect to a reference set of functions. Then, by taking into account also the Euclidean proximity between those discrete signals found to be adjustable, it becomes possible to obtain a respective transition network that can be not only used to study the properties and interrelationships of the involved discrete signals as underlain by the reference functions, but which also provide an interesting complex network theoretical model on itself, presenting a surprising diversity of topological features, including modular organization coexisting with more uniform portions, tails and handles, as well as hubs. Examples of the proposed concepts and methodologies are provided respectively with respect to three case examples involving power, sinusoidal and polynomial functions.

For a graph G=(V,E) with no isolated vertices, a set D⊆V is called a semipaired dominating set of G if (i) D is a dominating set of G , and (ii) D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G , and is denoted by γ pr2 (G) . The \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of G of cardinality γ pr2 (G) . In this paper, we initiate the algorithmic study of the \textsc{Minimum Semipaired Domination} problem. We show that the decision version of the \textsc{Minimum Semipaired Domination} problem is NP-complete for bipartite graphs and split graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs and trees. We also propose a 1+ln(2Δ+2) -approximation algorithm for the \textsc{Minimum Semipaired Domination} problem, where Δ denote the maximum degree of the graph and show that the \textsc{Minimum Semipaired Domination} problem cannot be approximated within (1−ϵ)ln|V| for any ϵ>0 unless NP ⊆ DTIME (|V | O(loglog|V|) ) .

We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from a general result about relational systems, which also implies the similar fact about high girth and high arboricity used in the proofs. These facts concern graphs and digraphs of high girth and low degree; we contrast them by considering acyclic colorings of tournaments (which have low girth and high degree). We prove that even though acyclic two-colorability of tournaments is known to be NP-complete, random acyclically r-colorable tournaments allow recovering an acyclic r-coloring in deterministic linear time, with high probablity.

Boolean networks are a general model of interacting entities, with applications to biological phenomena such as gene regulation. Attractors play a central role, and the schedule of entities update is a priori unknown. This article presents results on the computational complexity of problems related to the existence of update schedules such that some limit-cycle lengths are possible or not. We first prove that given a Boolean network updated in parallel, knowing whether it has at least one limit-cycle of length k is NP -complete. Adding an existential quantification on the block-sequential update schedule does not change the complexity class of the problem, but the following alternation brings us one level above in the polynomial hierarchy: given a Boolean network, knowing whether there exists a block-sequential update schedule such that it has no limit-cycle of length k is Σ P 2 -complete.

Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote by k -DLA the model where the particles move only in k possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid c and a sequence S of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site c when sequence S is realized. The second problem is Realization, where the input is a set of positions of the grid, P . The question is whether there exists a sequence S that realizes P , i.e. all particles of S exactly occupy the positions in P . Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is P-Complete for 2-DLA (thus for 3-DLA). Later, we show that Prediction can be solved much more efficiently for 1-DLA. In fact, we show that in that case the problem is NL-Complete. With respect to Realization, we show that restricted to 2-DLA the problem is in P, while in the 1-DLA case, the problem is in L.

Floating-point addition on a finite-precision machine is not associative, so not all mathematically equivalent summations are computationally equivalent. Making this assumption can lead to numerical error in computations. Proper ordering and parenthesizing is a low-overhead way of mitigating such error in a floating point summation. Ordered and parenthesized summations fall into equivalence classes. We describe these classes, and the parenthetic forms summations in these classes take. We provide summation-related interpretations for sequences known in other contexts, and give new recursive and closed formulas for sequences not previously related to summation. We also introduce a data structure that facilitates understanding of these objects, and use it to consider certain forms of summation used by default in widely used computer languages. Finally, we relate this data structure to other mathematical constructs from the fields of mathematical analysis and algorithmic analysis.

A graph G is weakly γ -closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that |N(u)∩N(v)|<γ . The weak closure γ(G) of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that G is weakly γ -closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s -plexes, are fixed-parameter tractable with respect to γ(G) . Moreover, we show that the problem of determining whether a weakly γ -closed graph G has a subgraph on at least k vertices that belongs to a graph class G which is closed under taking subgraphs admits a kernel with at most γ k 2 vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by γ+k where k is the solution size.

In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph G=(V,E) and a subset T of V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G , a temporal graph is represented by assigning a set of integer time-labels to every edge e of G , indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ∈N is given. The requirement that a vertex cannot be matched twice in any Δ -window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms.

A walk W between vertices u and v of a graph G is called a {\em tolled walk between u and v } if u , as well as v , has exactly one neighbour in W . A set S⊆V(G) is {\em toll convex} if the vertices contained in any tolled walk between two vertices of S are contained in S . The {\em toll convex hull of S } is the minimum toll convex set containing~ S . The {\em toll hull number of G } is the minimum cardinality of a set S such that the toll convex hull of S is V(G) . The main contribution of this work is an algorithm for computing the toll hull number of a general graph in polynomial time.