Electronic Notes in Theoretical Computer Science最新文献

This paper is inspired in the well known characterization of chordal graphs as the intersection graphs of subtrees of a tree. We consider families of induced trees of any graph and we prove that their recognition is NP-Complete. A consequence of this fact is that the concept of clique tree of chordal graphs cannot be widely generalized. Finally, we consider the fact that every graph is the intersection graph of induced trees of a bipartite graph and we characterize some classes that arise when we impose restrictions on the host bipartite graph.

A path partition $P$ of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition $P$ of D is defined as ${\sum }_{P\in P}\min \left\{\right|P_i|,k\}$. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacent vertices of V(D). Given a positive integer k, we denote by α_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that π_k(D) ≤ α_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial's Conjecture for arc-spine digraphs.

An oriented graph D is an orientation of a simple graph, i.e. a directed graph whose underlying graph is simple. A directed path from u to v with minimum number of arcs in D is an (u, v)-geodesic, for every u, v ∈ V(D). A set S ⊆ V(D) is (geodesically) convex if, for every u, v ∈ S, all the vertices in each (u, v)-geodesic and in each (v, u)-geodesic are in S. For every S ⊆ V(D) the (convex) hull of S is the smallest convex set containing S and it is denoted by [S]. A hull set of D is a set S ⊆ V(D) whose hull is V(D). The cardinality of a minimum hull set is the hull number of D and it is denoted by $\overrightarrow{hn}\left(D\right)$. A geodetic set of D is a set S ⊆ V(D) such that each vertex of D lies in an (u, v)-geodesic, for some u, v ∈ S. The cardinality of a minimum geodetic set is the geodetic number of D and it is denoted by $\overrightarrow{gn}\left(D\right)$.

In this work, we first present an upper bound for the hull number of oriented split graphs. Then, we turn our attention to the computational complexity of determining such parameters. We first show that computing $\overrightarrow{hn}\left(D\right)$ is NP-hard for partial cubes, a subclass of bipartite graphs, and that computing $\overrightarrow{gn}\left(D\right)$ is also NP-hard for directed acyclic graphs (DAG). Finally, we present a positive result by showing how to compute such parameters in polynomial time when the input graph is an oriented cactus.

Motivated by the significant advances in integer optimization in the past decade, Bertsimas and Shioda developed an integer optimization method to the classical statistical problem of classification in a multidimensional space, delivering a software package called CRIO (Classification and Regression via Integer Optimization). Following those ideas, we define a new classification problem, exploring its combinatorial aspects. That problem is defined on graphs using the geodesic convexity as an analogy of the Euclidean convexity in the multidimensional space. We denote such a problem by Geodesic Classification (GC) problem. We propose an integer programming formulation for the GC problem along with a branch-and-cut algorithm to solve it. Finally, we show computational experiments in order to evaluate the combinatorial optimization efficiency and classification accuracy of the proposed approach.

The ring-tree facility location problem is a generalization of the capacitated ring-tree problem in which additional cost and capacity related to facilities are considered. Applications of this problem arise in the strategic design of bi-level telecommunication networks. We investigate an extended integer programming formulation for the problem and different approaches to deal with the NP-hardness of the pricing problem that appears in a branch-and-price algorithm to solve it. Computational experiments show how heuristics and relaxations improved the performance of a branch-and-price algorithm.

A total coloring is equitable if the number of elements colored by any two distinct colors differs by at most one. The equitable total chromatic number of a graph $\left({\chi }_e^{″}\right)$ is the smallest integer for which the graph has an equitable total coloring. Wang (2002) conjectured that $\Delta +1\le {\chi }_e^{″}\le \Delta +2$. In 1994, Fu proved that there exist equitable (Δ + 2)-total colorings for all complete r-partite p-balanced graphs of odd order. For the even case, he determined that ${\chi }_e^{″}\le \Delta +3$. Silva, Dantas and Sasaki (2018) verified Wang's conjecture when G is a complete r-partite p-balanced graph, showing that ${\chi }_e^{″}=\Delta +1$ if G has odd order, and ${\chi }_e^{″}\le \Delta +2$ if G has even order. In this work we improve this bound by showing that ${\chi }_e^{″}=\Delta +1$ when G is a complete r-partite p-balanced graph with r ≥ 4 even and p even, and for r odd and p even.

A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of G is the set S_b(G) of integers k such that G has a b-coloring with k colors and b(G) = maxS_b(G) is the b-chromatic number of G. A graph is b-continous if $S_b\left(G\right)=\left[\chi \right(G),b(G\left)\right]\cap Z$. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph G with girth at least 7 contains the integers between 2χ(G) and b(G). This generalizes a previous result by Linhares-Sales and Silva (2017), and tells that graphs with girth at least 7 are, in a way, almost b-continuous.

A gap-[k]-vertex-labelling of a simple graph G = (V, E) is a pair (π, c_π) in which π : V (G) → {1, 2, ..., k} is an assignment of labels to the vertices of G and c_π : V (G) → {0, 1, ..., k} is a proper vertex-colouring of G such that, for every v ∈ V (G) of degree at least two, c_π(v) is induced by the largest difference, i.e. the largest gap, between the labels of its neighbours (cases where d(v) = 1 and d(v) = 0 are treated separately). Introduced in 2013 by A. Dehghan et al. [Dehghan, A., M. Sadeghi and A. Ahadi, Algorithmic complexity of proper labeling problems, Theoretical Computer Science 495 (2013), pp. 25–36.], they show that deciding whether a bipartite graph admits a gap-[2]-vertex-labelling is NP-complete and question the computational complexity of deciding whether cubic bipartite graphs admit such a labelling. In this work, we advance the study of the computational complexity for this class, proving that this problem remains NP-complete even when restricted to subcubic bipartite graphs.

Genome rearrangements are events that affect large portions of a genome. When using the rearrangement distance to compare two genomes, one wants to find a minimum cost sequence of rearrangements that transforms one into another. Since we represent genomes as permutations, we can reduce this problem to the problem of sorting a permutation with a minimum cost sequence of rearrangements. In the traditional approach, we consider that all rearrangements are equally likely to occur and we set a unitary cost for all rearrangements. However, there are two variations of the problem motivated by the observation that rearrangements involving large segments of a genome rarely occur. The first variation adds a restriction to the rearrangement's length. The second variation uses a cost function based on the rearrangement's length. In this work, we present approximation algorithms for five problems combining both variations, that is, problems with a length-limit restriction and a cost function based on the rearrangement's length.

In this paper, we consider a coloring as a function that assigns a color to a vertex, regardless of the color of its neighbors. The Convex Recoloring Problem finds the minimum number of recolored vertices needed to turn a coloring convex, that is, every set formed by all the vertices with the same color induces a connected subgraph. The problem is most commonly studied considering trees due to its origins in the study of phylogenetic trees, but in this paper, we focus on general graphs and propose a GRASP heuristic to solve the problem. We present computational experiments for our heuristic and compare it to an Integer Linear Programming model from the literature. In these experiments, the GRASP algorithm recolored a similar number of vertices than the model from the literature, and used considerably less time. We also introduce a set of benchmark instances for the problem.