European Journal of Operational Research 262 (2017) 464–478 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Discrete Optimization Locating names on vertices of a transaction network David Alcaide-López-de-Pablo * , Joaquín Sicilia , Miguel Á. González-Sierra Departamento de Matemáticas, Estadística e Investigación Operativa, Universidad de La Laguna, San Cristóbal de La Laguna, Tenerife, Spain a r t i c l e i n f o Article history: Received 22 January 2015 Accepted 5 April 2017 Available online 20 April 2017 Keywords: Assignment Graph theory Integer programming Heuristic algorithms a b s t r a c t This paper deals with the problem of identifying names of unknown subjects located on vertices of a net- work which is previously established among subjects. The only available information is the knowledge of the names of some subjects in the network, and certain records of previous observations of transactions between all pairs of subjects. Such records offer us information about the frequency or intensity of these transactions. The aim is to find the more suitable identification of the unknown subjects taking into ac- count the information about the frequency of transactions among subjects. It is proved that the problem is NP-hard. A heuristic approach is proposed for solving it and its performance is numerically illustrated. © 2017 Elsevier B.V. All rights reserved. 1. Introduction Harary, Morgana, and Simeone (1997) studied the problem of filling the missing names of towns in a map. They formulated their problem with a graph theoretic approach. In their paper, they considered that the map showed only the names of some, but not all, towns in a region. For each town, the names of all neigh- bouring towns were previously known. Taking into account that information, they provided a procedure to reconstruct the missing names in the cases in which such reconstruction is possible. They also dealt with this problem for arbitrary undirected graphs, using a known list of neighbourhoods for the names. We study here a generalisation of the problem analysed by Harary et al. (1997), when the neighbours of each town are not known with complete precision. Our approach also works with arbitrary undirected graphs, but the neighbourhood among every pair of names is assessed by a value between zero and one which represents the neighbourhood degree between them. Even more, this point of view also allows us to study other problems in which there are relationships among subjects. These relationships can be modelled by an undirected graph, but the names of the nodes of such graph are not completely known. The problem would consist of determining, in the most suitable way, the names of all the subjects. We can refer to it as the collaboration network problem. In addition, we extend the undirected graph model to a directed graph one. This directed graph model represents better the situa- tions in which the relationships are transactions between subjects * Corresponding author. E-mail addresses: dalcaide@ull.es (D. Alcaide-López-de-Pablo), jsicilia@ull.es (J. Sicilia), magsierr@ull.es (M.Á. González-Sierra). and the frequency or value of such transactions is not symmetrical. We can mention it as the transaction network problem. Some instances of such problems could appear in the following real situations: (i) Let us consider a set of jobs that must be performed by a set of workers in any enterprise, industry or organisation. Each job must be performed by only one worker and any worker may per- form any job. No more than one job is entrusted to each worker. For performing a pair of jobs some shared resources are neces- sary (for example tools, machines, computers, devices, physical space, etc.), and therefore some type of compatibility or collabo- ration could be required between the workers in charge of these jobs. This collaboration structure is shown by an undirected graph, where the nodes are jobs, and two nodes are adjacent if their as- sociated jobs require share some resources. As a result, the work- ers assigned to such jobs must collaborate. This is the reason be- cause we are calling this network a collaboration network: the jobs (vertices) which demand shared resources are linked in the net- work and, consequently, the workers (labels) assigned to linked jobs must collaborate among them. Consider also an affinity re- lationship among workers. The affinity degree between any pair of workers is measured by a numerical value that we can normalise to a value between zero and one. If this last value is close to one, it means that there exists better understanding and easier collab- oration between the workers. We are interested to assign workers with good collaboration among them to the jobs which demand shared resources, and workers with no so easy collaboration to jobs that do not need shared resources. The problem would be to find the most suitable allocation of the workers to the jobs in such a way the total affinity/collaboration among workers assigned to linked jobs is maximised. This fact will be advantageous for the http://dx.doi.org/10.1016/j.ejor.2017.04.011 0377-2217/© 2017 Elsevier B.V. All rights reserved.