arXiv subjects

Graciela Nasini

Publications and source records attributed to Graciela Nasini.

15 recordsLinked to original sources

arXiv2023

Solving the List Coloring Problem through a Branch-and-Price algorithm

In this work, we present a branch-and-price algorithm to solve the weighted version of the List Coloring Problem, based on a vertex cover formulation by stable sets. This problem is interesting for its applications and also for the many other problems that it generalizes, including the well-known Graph Coloring Problem. With the introduction of the concept of indistinguishable colors, some theoretical results are presented which are later incorporated into the algorithm. We propose two branching strategies based on others for the Graph Coloring Problem, the first is an adaptation of the one used by Mehrotra and Trick in their pioneering branch-and-price algorithm, and the other is inspired by the one used by Méndez-Díaz and Zabala in their branch-and-cut algorithm. The rich structure of this problem makes both branching strategies robust. Extended computation experimentation on a wide variety of instances shows the effectiveness of this approach and evidences the different behaviors that the algorithm can have according to the structure of each type of instance.

cs.DS↗

arXiv2019

Circuits and circulant minors

Circulant contraction minors play a key role for characterizing ideal circular matrices in terms of minimally non ideal structures. In this article we prove necessary and sufficient conditions for a circular matrix $A$ to have circulant contraction minors in terms of circuits in a digraph associated with $A$. In the particular case when $A$ itself is a circulant matrix, our result provides an alternative characterization to the one previously known from the literature.

math.CO↗

arXiv2018

A Branch and Price Algorithm for List Coloring Problem

Coloring problems in graphs have been used to model a wide range of real applications. In particular, the List Coloring Problem generalizes the well-known Graph Coloring Problem for which many exact algorithms have been developed. In this work, we present a Branch-and-Price algorithm for the weighted version of the List Coloring Problem, based on the one developed by Mehrotra and Trick (1996) for the Graph Coloring Problem. This version considers non-negative weights associated to each color and it is required to assign a color to each vertex from predetermined lists in such a way the sum of weights of the assigned colors is minimum. Computational experiments show the good performance of our approach, being able to comfortably solve instances whose graphs have up to seventy vertices. These experiences also bring out that the hardness of the instances of the List Coloring Problem does not seem to depend only on quantitative parameters such as the size of the graph, its density, and the size of list of colors, but also on the distribution of colors present in the lists.

cs.DM↗

arXiv2018

Grundy dominating sequences on $X$-join product

In this paper we study the Grundy domination number on the $X$-join product $G\hookleftarrow \mathcal R$ of a graph $G$ and a family of graphs $\mathcal R=\{G_v: v\in V(G)\}$. The results led us to extend the few known families of graphs where this parameter can be efficiently computed. We prove that if, for all $v\in V(G)$, the Grundy domination number of $G_v$ is given, and $G$ is a power of a cycle, a power of a path, or a split graph, computing the Grundy domination number of $G\hookleftarrow \mathcal R$ can be done in polynomial time. In particular, the results for power of cycles and paths are derived from a polynomial reduction to the Maximum Weight Independent Set problem on these graphs. As a consequence, we derive closed formulas to compute the Grundy domination number of the lexicographic product $G\circ H$ when $G$ is a power of a cycle, a power of a path or a split graph, generalizing the results on cycles and paths given by Bresar et al. in 2016. Moreover, the results on the $X$-join product when $G$ is a split graph also provide polynomial-time algorithms to compute the Grundy domination number for $(q,q-4)$ graphs, partner limited graphs and extended $P_4$-laden graphs, graph classes which are high in the hierarchy of few $P_4$'s graphs.

math.CO↗

arXiv2018

On dominating set polyhedra of circular interval graphs

Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. take advantage of this relationship to propose a complete linear description of the stable set polytope on circular interval graphs. In this paper we follow similar ideas to obtain a complete description of the dominating set polytope on the same class of graphs. As in the packing case, our results are established for a larger class of covering polyhedra of the form Q*(A,b):= conv{x \in \Z^n_+: Ax >= b}, with A a circular matrix and b an integer vector. These results also provide linear descriptions of polyhedra associated with several variantsof the dominating set problem on circular interval graphs.

math.CO↗

arXiv2016

Lovász-Schrijver PSD-operator on Claw-Free Graphs

The subject of this work is the study of $\LS_+$-perfect graphs defined as those graphs $G$ for which the stable set polytope $\stab(G)$ is achieved in one iteration of Lovász-Schrijver PSD-operator $\LS_+$, applied to its edge relaxation $\estab(G)$. In particular, we look for a polyhedral relaxation of $\stab(G)$ that coincides with $\LS_+(\estab(G))$ and $\stab(G)$ if and only if $G$ is $\LS_+$-perfect. An according conjecture has been recently formulated ($\LS_+$-Perfect Graph Conjecture); here we verify it for the well-studied class of claw-free graphs.

math.CO↗

arXiv2016

Total dominating sequences in trees, split graphs, and under modular decomposition

A sequence of vertices in a graph $G$ with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of $G$ are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, $γ_{\rm gr}^t(G)$, of $G$, as introduced in [B. Brešar, M.A. Henning, and D. F. Rall, Total dominating sequences in graphs, Discrete Math. 339 (2016), 1165--1676]. In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary tree $T$ is presented, based on the formula $γ_{\rm gr}^t(T)=2τ(T)$, where $τ(T)$ is the vertex cover number of $T$. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of $P_4$-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs $G_k$ such that $γ_{\rm gr}^t(G_k)=k$, for any $k\in{\mathbb{Z}^+}\setminus\{1,3\}$, and showing that there are no graphs $G$ with $γ_{\rm gr}^t(G)\in \{1,3\}$. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to $k$.

math.CO↗

arXiv2015

On the Packing Chromatic Number on Hamming Graphs and General Graphs

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ needed to proper color the vertices of $G$ in such a way the distance between any two vertices having color $i$ be at least $i+1$. We obtain $χ_ρ(H_{q,m})$ for $m=3$, where $H_{q,m}$ is the Hamming graph of words of length $m$ and alphabet with $q$ symbols, and tabulate bounds of them for $m \geq 4$ up to 10000 vertices. We also give a polynomial reduction from the problem of finding $χ_ρ(G)$ to the Maximum Stable Set problem.

cs.DM↗

arXiv2014

The Identifying Code problem on $P_4$-tidy graphs

In this paper we first analyze the behaviour of the identifying code number under union and join operations in graphs. This study forced us to analyze three new parameters related to identifying codes, dominating sets and total dominating sets of the complementary graph. We obtain closed formulas for these parameters on spider and quasi-spider graphs. These results easily derive in a linear dynamic programming-based algorithm for the Identifying Code problem for $P_4$-tidy graphs.

math.CO↗

arXiv2014

A tabu search heuristic for the Equitable Coloring Problem

The Equitable Coloring Problem is a variant of the Graph Coloring Problem where the sizes of two arbitrary color classes differ in at most one unit. This additional condition, called equity constraints, arises naturally in several applications. Due to the hardness of the problem, current exact algorithms can not solve large-sized instances. Such instances must be addressed only via heuristic methods. In this paper we present a tabu search heuristic for the Equitable Coloring Problem. This algorithm is an adaptation of the dynamic TabuCol version of Galinier and Hao. In order to satisfy equity constraints, new local search criteria are given. Computational experiments are carried out in order to find the best combination of parameters involved in the dynamic tenure of the heuristic. Finally, we show the good performance of our heuristic over known benchmark instances.

cs.DM↗

arXiv2014

A DSATUR-based algorithm for the Equitable Coloring Problem

This paper describes a new exact algorithm for the Equitable Coloring Problem, a coloring problem where the sizes of two arbitrary color classes differ in at most one unit. Based on the well known DSatur algorithm for the classic Coloring Problem, a new pruning criterion arising from equity constraints is proposed and analyzed. The good performance of the algorithm is shown through computational experiments over random and benchmark instances.

cs.DM↗

arXiv2014

An exact DSatur-based algorithm for the Equitable Coloring Problem

This paper describes an exact algorithm for the Equitable Coloring Problem, based on the well known DSatur algorithm for the classic Coloring Problem with new pruning rules specifically derived from the equity constraint. Computational experiences show that our algorithm is competitive with those known in literature.

cs.DM↗

arXiv2012

Some advances on the set covering polyhedron of circulant matrices

Working on the set covering polyhedron of consecutive ones circulant matrices, Argiroffo and Bianchi found a class of facet defining inequalities, induced by a particular family of circulant minors. In this work we extend these results to inequalities associated with every circulant minor. We also obtain polynomial separation algorithms for particular classes of such inequalities.

math.CO↗

arXiv2011

A polyhedral approach for the Equitable Coloring Problem

In this work we study the polytope associated with a 0,1-integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence that shows the efficacy of these inequalities used in a cutting-plane algorithm.

cs.DM↗

arXiv2011

Polyhedral results for the Equitable Coloring Problem

In this work we study the polytope associated with a 0/1 integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence of the effectiveness of including these inequalities as cuts in a Branch & Cut algorithm.

cs.DM↗