Electronic Notes in Discrete Mathematics最新文献

In the logistics industry, not only cartons but also polybags are used to pack goods for delivery. We study a single polybag loading problem, which is faced by many e-retailers and courier companies. In the problem, a set of rotatable boxes and a two-dimensional rectangular polybag are given. Because the polybag is flexible, the three-dimensional space inside the polybag can be variable when the boxes are loaded. The problem is to choose a subset of the boxes and pack them orthogonally into the polybag so as to maximize the space utilization. We introduce a mixed integer programming formulation and propose a tree search heuristic for the problem. Methods are evaluated by computational experiments on randomly generated data.

In this work, we consider the graph sandwich problem for a property Π, a decision problem proposed by Golumbic, Kaplan, and Shamir as follows: given two graphs G¹ = (V, E¹) and G² = (V, E²), the question is whether there exists a graph G = (V, E) such that E¹ ⊆ E ⊆ E² and G satisfies Π. For many graph classes, this problem was settled to be NP-complete. For this reason, a different kind of approach has been proposed: instead of focusing only on property Π, we can also require special properties ${\Pi }^i$ for the input graphs Gⁱ, i = 1, 2, resulting in a problem that generalizes graph sandwich problems, called graph sandwich problems with boundary conditions. This problem is denoted by a triple (${\Pi }^1$, Π, ${\Pi }^2$)-sp. We deal with the property Π of being a chordal-(2,1)-graph when it is given that G² is a graph of a class for which there is a polynomial bound on the number of cliques, say pnc. chordal-(2,1)-sp is known to be NP-complete but, when requiring G² to be of a class in pnc, we prove that (*, chordal-(2,1), pnc)-sp is polynomially time solvable.

We study the problem of designing walking school bus lines (Pedibus) limiting the deviation with respect to the shortest path for each child, with the objective of minimizing the number of accompanying persons and the perceived risk of the selected trajectories. The problem is formulated using a path model and a column generation approach is proposed. Computational experiments compare the lower bounds and the solutions of the proposed approach with the arc model and a simple heuristic proposed in a previous work.

Genome rearrangement problems aim at finding the minimum number of mutational events required to transform a genome into another. Studying such problems from a combinatorial point of view has been proved to be useful for the reconstruction of phylogenetic trees. In this work we focus on the relations between genome rearrangement and graph convexity in the following way. Graph convexity problems deal with input sets of vertices and try to understand properties on the closure of such inputs. In turn, the concept of closure is useful for studies on genome rearrangement by suggesting mechanisms to reduce the genomic search space. Computational complexity studies are widely developed in graph convexity problems, since there are several types of convexities, each capturing a distinct way of defining what is meant by “closure”. In this regard, considering the Hamming distance of strings, we solve the following problems: decide whether a given set is convex; compute the interval and the convex hull of a given set; and determine the convexity number of a given graph. All such problems are solved for the geodesic and monophonic convexities. On the other hand, for the Cayley distance of permutations, we study convexity of sets and interval determination of sets, in the context of the geodesic convexity.

Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set S is a vertex separator for vertices a and b if the removal of S of the graph separates a and b into distinct connected components. A graph G is chordal if and only if every minimal vertex separator is a clique. We study subclasses of chordal graphs defined by restrictions imposed on the intersections of its minimal separator cliques. Our goal is to characterize them by forbidden induced subgraphs. Some of these classes have already been studied such as chordal graphs in which two minimal separators have no empty intersection if and only if they are equal. Those graphs are known as strictly chordal graphs and they were first introduced as block duplicate graphs by Golumbic and Peled [Golumbic, M. C. and Peled, U. N., Block duplicate graphs and a hierarchy of chordal graphs, Discrete Applied Mathematics, 124 (2002) 67–71], they were also considered in [Kennedy, W., “Strictly chordal graphs and phylogenetic roots”, Master Thesis, University of Alberta, 2005] and [De Caria, P. and Gutiérrez, M., On basic chordal graphs and some of its subclasses, Discrete Applied Mathematics, 210 (2016) 261–276], showing that strictly chordal graphs are exactly the (gem, dart)-free graphs.

In this paper we address a Pickup and Delivery Vehicle Routing Problem where the demands of both pickup and delivery nodes can be split between several vehicles. We investigate a new formulation where the route of each vehicle is decomposed into a sequence of simpler substructures called clusters, mitigating the combinatorial explosion of feasible solutions. We implement a branch-and-price algorithm exploiting column generation procedures that dynamically generates clusters to obtain improved dual bounds, and ad hoc branching strategies to achieve integrality.

Drones are more and more important in emergency response scenarios, where they can be equipped with wireless access points and deployed to configure a recovery ad-hoc flying network, e.g., to interconnect users positioned in a disaster area. Communication between users is allowed by means of multi-hop paths including one or more relay drones. An interesting optimization problem asks for determining the position of a minimum number of drones in order to guarantee that all the required communication paths can be established. Wireless channels are used and interference plays an important role: we propose an interference model and we embed it in a Mixed Integer Linear Programming formulation. Preliminary computational results are presented, showing the proposed model is able to solve instances of medium size in order of minutes.

We study minor related row family inequalities for the set covering polyhedron of circular matrices. We address the issue of generating these inequalities via the Chvátal-Gomory procedure and establish a general upper bound for their Chvátal-rank. Moreover, we provide a construction to obtain facets with arbitrarily large coefficients and examples of facets having Chvátal-rank strictly larger than one.

We consider the problem of finding microgrids in a network. We mainly focus on the complexity aspects related to the different variants of this problem⁵.

We consider the retrieval of articles from a warehouse to fulfill customer orders of a large e-commerce business. The problem consists of an order batching phase, where orders are grouped together to be processed by one picking process, and a routing phase where the article locations of one batch are sequenced in a route of minimal length. Our application is characterized by several non-standard features concerning the warehouse structure as well as the storage strategy. We develope a heuristic solution algorithm based on a fairly general graph model.