Total Dual Integrality in Some Facility Location Problems

Facility location, arising in a rich variety of applications, has been studied extensively in the fields of operations research and computer science. In this note we consider the classical uncapacitated facility location problem and its “prize-collecting” variant introduced by Bäıou and Barahona, and show that the linear systems associated with these problems are totally dual integral (TDI) if and only if the input graphs contain no certain type of odd cycles. As corollaries, we get structural characterizations of two min-max relations on facility location. Our results strengthen the integrality theorems on facility location polytopes proved by Bäıou and Barahona; our proofs lead to combinatorial polynomial-time algorithms for the facility location problems in our consideration. ∗Supported in part by NSF of China under Grant 10771209. †Supported in part by the Research Grants Council of Hong Kong and Seed Funding for Basic Research of HKU. ‡Corresponding author. E-mail: [email protected].