Precedence constrained generalized traveling salesman problem: Polyhedral study, formulations, and branch-and-cut algorithm

Keywords Travelling salesman; Integer programming; Branch-and-cut algorithm; Facet-inducing inequalities; Polyhedral structure Highlights * We address the Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP). * We propose novel families of valid inequalities for the PCGTSP. * We prove the first facet-inducing inequalities for the PCGTSP. * We introduce MILP-models (formulations) and Branch-and-Cut algorithm for the PCGTSP. * We prove high performance of proposed models and algorithm by numerical evaluation. Abstract The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) is an extension of two well-known combinatorial optimization problems -- the Generalized Traveling Salesman Problem (GTSP) and the Precedence Constrained Asymmetric Traveling Salesman Problem (PCATSP), whose path version is known as the Sequential Ordering Problem (SOP). Similarly to the classic GTSP, the goal of the PCGTSP, for a given input digraph and partition of its node set into clusters, is to find a minimum cost cyclic route (tour) visiting each cluster in a single node. In addition, as in the PCATSP, feasible tours are restricted to visit the clusters with respect to the given partial order. Unlike the GTSP and SOP, to the best of our knowledge, the PCGTSP still remain to be weakly studied both in terms of polyhedral theory and algorithms. In this paper, for the first time for the PCGTSP, we propose several families of valid inequalities, establish dimension of the PCGTS polytope and prove sufficient conditions ensuring that the extended Balas' [pi]- and [sigma]-inequalities become facet-inducing. Relying on these theoretical results and evolving the state-of-the-art algorithmic approaches for the PCATSP and SOP, we introduce a family of MILP-models (formulations) and several variants of the branch-and-cut algorithm for the PCGTSP. We prove their high performance in a competitive numerical evaluation against the public benchmark library PCGTSPLIB, a known adaptation of the classic SOPLIB to the problem in question. Author Affiliation: (a) Kedge Business School, 680 Cours de la Liberation, Talence, France (b) Inria Centre at the University of Bordeaux, 200 avenue de la Vieille Tour, Talence, France (c) Ivannikov Institute for System Programming of RAS, Moscow, Russia * Corresponding author. Article History: Received 18 April 2022; Accepted 21 January 2023 (footnote)[white star] Polyhedral study, formulations, and Branch-and-Cut for PCGTSP. Byline: Daniil Khachai [daniil.khachai@kedgebs.com] (*,a), Ruslan Sadykov [ruslan.sadykov@inria.fr] (b), Olga Battaia [olga.battaia@kedgebs.com] (a), Michael Khachay [mkhachay@imm.uran.ru] (c)