Facility Location Set of facilities F Set of clients D Facility Opening cost f i Connection cost of facility i to client j c ij Goal: Open a collection of facilities and connect to clients and minimize total cost Many figures, text from David Williamson’s slides on primal-dual algorithms

Facility Location Set of facilities F Set of clients D Facility Opening cost f i Connection cost of facility i to client j c ij Goal: Open a collection of facilities and connect to clients and minimize total cost

IP Formulation

Dual problem and algorithm

Layout with distances c ij

Layout with facility costs f i

Increasing all v j,no neighbors

First Neighbor

Increasing all v j, w 11 All clients now have neighbors

Client 1 has second neighbor

Facility 1 and 2 maxed out Facilities 1 and 2 added to A Clients 1 and 2 stop growing

Client 3 stops growing Neighbor with facility 2 and stops

Facility 3 added to A

Choosing Facilities Consider facilities i in A in order inequalities became tight. Open facility i if no neighboring facility of a neighboring client is open

Assigning Clients Assign client j to open neighbor, if any Otherwise to the first opened neighbor (fac) of a neighbor (client) of a neighbor (facility). Why must such a neighbor exist? 1 3

Analysis of Connected Neighbors Consider any open facilitiy i such as facility 3 All client neighbors are connected to i Each c ij = v j – w ij Thus Σc ij = Σv j – Σw ij = Σv j – f i Thus total cost of open facilities and connected neighbors is Σv j 3

Analysis of Distant Clients Consider any client j such as client 2 connected to a distant facility Argue c ij ≤ 3 v j Thus Σc ij ≤ 3Σv j Total approximation ratio of 3