Abstract: Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by in(cc)(G), is the minimum cardinality of a set S subset of V (G) such that every vertex w is an element of E V (G) \ S has two distinct neighbors u, v is an element of S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S. In this work, first we provide bounds on in(cc) (G) and its relations to other graph convexity parameters, and explore its behaviour on grids. Then, we present some hardness results by showing that deciding whetherin(cc) (G) <= k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtain that in(cc) (G) cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = NP). On the positive side, we present polynomial-time algorithms to compute in(cc) (G) for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q – 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G. Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.

Abstract: A set of autonomous robots have to collaborate in order to accomplish a common task in a ring-topology where neither nodes nor edges are labeled (that is, the ring is anonymous). We present a unified approach to solve three important problems: the exclusive perpetual exploration, the exclusive perpetual clearing, and the gathering problems. In the first problem, each robot aims at visiting each node infinitely often while avoiding that two robots occupy a same node (exclusivity property); in exclusive perpetual clearing (also known as graph searching), the team of robots aims at clearing the whole ring infinitely often (an edge is cleared if it is traversed by a robot or if both its endpoints are occupied); and in the gathering problem, all robots must eventually occupy the same node. We investigate these tasks in the Look-Compute-Move model where the robots cannot communicate but can perceive the positions of other robots. Each robot is equipped with visibility sensors and motion actuators, and it operates in asynchronous cycles. In each cycle, a robot takes a snapshot of the current global configuration (Look), then, based on the perceived configuration, takes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case it eventually moves to this neighbor (Move). Moreover, robots are endowed with very weak capabilities. Namely, they are anonymous, asynchronous, oblivious, uniform (execute the same algorithm) and have no common sense of orientation. In this setting, we devise algorithms that, starting from an exclusive and rigid (i.e. aperiodic and asymmetric) configuration, solve the three above problems in anonymous ring-topologies.

Abstract: Given a vertex-weighted tree T, the split of an edge em T is the minimum over the weights of the two trees obtained by removing e from T, where the weight of a tree is the sum of weights of its vertices. Given a set of weighted vertices V and a multiset of integers s, we consider the problem of constructing a tree on V whose splits correspond to s. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be at most 4. We show that the problem is strongly NP-complete when T is required to be a path, the problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be at most 3, and it remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3. We also design polynomial time algorithms for the variant where T is required to be a path and the number of distinct vertex weights is constant, and the variant where all vertices have unit weight and T has a constant number of leaves. The latter algorithm is not only polynomial when the number of leaves, k, is a constant, but also is a fixed-parameter algorithm for parameter k. Finally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm. The considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong relation between structural invariants of the particles, such as the Wiener index and, less directly, the problem under consideration here, and physico-chemical properties of the substance. (C) 2014 Elsevier B.V. All rights reserved.

Abstract: We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP subset of coNP/poly), for planar graphs Guo and Niedermeier [ICALP'08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS'11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this problem? In this paper we answer this question in the negative: we show a 11/3 k-vertex kernel for CONNECTED VERTEX COVER in planar graphs. We believe that this result will motivate further study in the search for an optimal kernel. In our analysis, we show an extension of a theorem of Nishizeki and Baybars [Takao Nishizeki, Ilker Baybars, Lower bounds on the cardinality of the maximum matchings of planar graphs, Discrete Mathematics 28 (3) (1979) 255-267] which might be of independent interest: every planar graph with n(>= 3) vertices of degree at least 3 contains a matching of cardinality at least n(>= 3)/3. (C) 2012 Elsevier B.V. All rights reserved.

Abstract: This study introduces the Territory Design for the Multi-Period Vehicle Routing Problem with Time Windows (TD-MPVRPTW) problem, motivated by a real-world application at a food company's distribution center. This problem deals with the design of contiguous and compact territories for delivery of orders from a depot to a set of customers, with time windows, over a multi-period planning horizon. Customers and their demands vary over time. The problem is modeled as a mixed-integer linear program (MILP) and solved by a proposed heuristic. The heuristic solutions are compared with the proposed MILP solutions on a set of small artificial instances and the food company's solutions on a set of real-world instances. Computational results show that the proposed algorithm can yield high-quality solutions within moderate running times. A methodology is proposed in which the territories computed by the proposed heuristic on the past demand of one month are used for the operational routing during the following month, in which the demand is known only one day in advance. An evaluation shows that the territories obtained with our methodology would have led to levels of service significantly better than the ones that were experienced by the company, using a significantly lower number of vehicles to execute the deliveries.

Abstract: Vehicle Routing Problems (VRP) comprise many variants obtained by adding to the original problem constraints representing diverse system characteristics. Different variants are widely studied in the literature; however, the impact that these constraints have on the structure of the search space associated with the problem is unknown, and so is their influence on the performance of search algorithms used to solve it. This article explores how assignation constraints (such as a limited vehicle capacity) impact VRP by disturbing the network structure defined by the solution space and the local operators in use. This research focuses on Fitness Landscape Analysis for the multiple Traveling Salesman Problem (m-TSP) and Capacitated VRP (CVRP). We propose a new Fitness Landscape Analysis measure that provides valuable information to characterize the fitness landscape's structure under specific scenarios and obtain several relationships between the fitness landscape's structure and the algorithmic performance.

Abstract: Efficient algorithms for computing routing tables should take advantage of particular properties arising in large scale networks. Two of them are of special interest: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of k-chordal graphs, i.e., graphs with no induced cycles of length more than k. In the class of k-chordal graphs, our routing scheme achieves an additive stretch of at most k – 1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k – 1. In order to compute the routing tables of any n-node graph with diameter D we propose a distributed algorithm which uses O(log n)-bit messages and takes O(D) time. The corresponding routing scheme achieves the stretch of k – 1 on k-chordal graphs. We then propose a routing scheme that achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, distributed computation of routing tables takes O(min{Delta D, n}) time, where Delta is the maximum degree of the graph. Our routing schemes use addresses of size log n bits and local memory of size 2(d-1) log n bits per node of degree d. (c) 2012 Elsevier B.V. All rights reserved.