Gotta (efficiently) catch them all: Pokémon GO meets orienteering problems.

*(English)*Zbl 1374.90395Summary: In this paper, a new routing problem, referred to as the Generalized Clustered Orienteering Problem (GCOP), is studied. The problem is motivated by the mobile phone game Pokémon GO, an augmented reality game for mobile devices holding a record-breaking reception: within the first month of its release, more than 100 million users have installed the game on their devices. The game’s immense popularity has spawned several side businesses, including taxi-tours visiting locations where the game can be played, as well as companies offering to play the game for users during times when they cannot. Further applications arise in typical operative transportation problems that seek for tours that are both time-effective and profitable. Besides the typical traveling distances, in the GCOP we also have prizes or revenues associated with the nodes. Additionally, we are given with \(K\) node subsets (clusters) and a budget \(B\) for the length of the tour. The optimization task is to find a tour that maximizes the total collected prize while ensuring that (i) at least one node of each cluster is visited, and (ii) the total distance of the tour does not exceed the budget B. In order to solve the GCOP to optimality, a polynomial-sized Mixed-Integer Linear Programming (MIP) formulation and an exponential-sized MIP formulation are presented. While the first formulation is tackled by a state-of-the-art branch-and-bound (B&B) algorithm, the second formulation is approached by a specially tailored branch-and-cut (B&C) framework; moreover, the proposed B&C is further enhanced with valid inequalities, a lifting procedure for strengthening inequalities, as well as initialization and primal heuristics. The computational performance of the proposed approaches is assessed in an extensive computational study, using real-world instances that combine crowd-sourced data associated with the Pokémon GO game with street maps of three European cities, as well as instances derived from the TSPLIB testbed. The obtained results show that the B&C approach (i) largely outperforms the B&B algorithm, and that (ii) it is very effective for providing optimal or nearly-optimal solutions within reasonable running times for both sets of instances.

##### MSC:

90C35 | Programming involving graphs or networks |

90C10 | Integer programming |

90C27 | Combinatorial optimization |

90C57 | Polyhedral combinatorics, branch-and-bound, branch-and-cut |

##### Keywords:

combinatorial optimization; orienteering problem; generalized traveling salesman problem; integer programming; branch-and-cut
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\textit{E. Álvarez-Miranda} et al., Eur. J. Oper. Res. 265, No. 2, 779--794 (2018; Zbl 1374.90395)

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