Intl. Trans. in Op. Res. 25 (2018) 295–318 DOI: 10.1111/itor.12444 INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH Complexity among combinatorial problems from epidemics Juan Piccini, Franco Robledo and Pablo Romero Laboratorio de Probabilidad y Estad´ ıstica, Facultad de Ingenier´ ıa, Universidad de la Rep´ ublica, Julio Herrera y Reissig 565, Montevideo, Uruguay E-mail: piccini@fing.edu.uy [Piccini]; frobledo@fing.edu.uy [Robledo]; promero@fing.edu.uy [Romero] Received 5 May 2016; received in revised form 31 March 2017; accepted 27 June 2017 Abstract A cornerstone in epidemic modeling is the classical susceptible–infected–removed model, or SIR. In this model, individuals are divided into three classes: susceptible (those who can be infected), infected, and removed (those who suffered the infection and recovered, gaining immunity from further contact with infected individuals). Transitions S → I → R occur at constant rates γ S ,γ I . The SIR model is both simple and useful to understand cascading failures in a network. Nevertheless, a shortcoming is the unrealistic assumption of random contacts in a fully mixed large population. More realistic models are available from authoritative literature in the field. They consider a graph and an epidemic spread governed by probabilistic rules. In this paper, a combinatorial optimization problem is introduced using graph-theoretic terminology, inspired by an extremal analysis of epidemic modeling. The contributions are threefold. First, a general node immunization problem is defined for node immunization under budget requirements, using probabilistic networks. The goal is to minimize the expected number of deaths under a particular choice of nodes in the system to be immunized. In the second stage, a highly virulent environment leads to a purely combinatorial problem without probabilistic law, called the graph fragmentation problem (GFP). We prove the corresponding decision version for the GFP belongs to the class of NP -complete problems. As a corollary, SIR-based models also belong to this set. Third, a GRASP (greedy randomized adaptive search procedure) heuristic enriched with a path-relinking post-optimization phase is developed for the GFP. Finally, an experimental analysis is carried out under graphs taken from real-life applications. Keywords: combinatorial optimization problem; graph fragmentation problem; computational complexity 1. Introduction The most valuable aspect of the susceptible–infected–removed (SIR) model is its simplicity: closed formulas are met, an epidemic spread can easily be carried out on a computer, and it connects deterministic and stochastic models in an elegant manner. For these reasons, the SIR model is the starting point in teaching and understanding an epidemic process. However, it assumes a full-mixed large population with random contacts, and identical contagion probabilities for all the individuals. C  2017 The Authors. International Transactions in Operational Research C  2017 International Federation of Operational Research Societies Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148, USA.