LP Solvable Location Problems on Networks

Given a network, the facility location problem consists of locating a set of sites on the network to serve existing clients so as to minimize its total cost. Special cases include the simple plant location problem (SPLP), bank account location, p-facility location, and p-median problem. We characterize families of networks on which all basic feasible solution from simple linear programming (LP) relaxations are integral. For all these location problems, we present a necessary and sufficient condition for which their basic LP solutions are integral. For the SPLP, we show that the basic LP solutions are integral if and only if the network contains no cycles of length 2 (mod 4), or equivalently, the graph is obtainable from an even ear composition. Finally, we study polyhedral structures of SPLP and characterize a class of facet inducing inequalities induced by odd cycles.