Hybrid meta-heuristics with VNS and exact methods: application to large unconditional and conditional vertex p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\

Large-scale unconditional and conditional vertex $$p$$p-centre problems are solved using two meta-heuristics. One is based on a three-stage approach whereas the other relies on a guided multi-start principle. Both methods incorporate Variable Neighbourhood Search, exact method, and aggregation techniques. The methods are assessed on the TSP dataset which consist of up to 71,009 demand points with $$p$$p varying from 5 to 100. To the best of our knowledge, these are the largest instances solved for unconditional and conditional vertex $$p$$p-centre problems. The two proposed meta-heuristics yield competitive results for both classes of problems.