On collection depots location problems Binay K Bhattacharya

On collection depots location problems Binay K Bhattacharya School of Computing Science Simon Fraser University (Joint work with Robert Benkoczi, Sandip Das, Jeff Sember and Arie Tamir)

What is facility locating? Facility provides some kind of service. Client needs this kind of service. ‘GOOD’ SERVICE

Where should the radio tower be located? • Tower may be positioned anywhere – (continuous) • Tower may be positioned in five available slots – (discrete) • Tower may be located on the roadside – (network)

Distance metric • Minkowski distance (Euclidean, Manhattan) • Network distance

Classical k-center problem Given a universe U (where facilities can be located), a set of clients C, a metric d, and a positive integer k, a k-center of C is a set of k points F of the universe U that minimizes maxjεC {miniεF dij}.

Classical k-median problem Given a universe U, a set of points C, a metric d, and a positive integer k, a k-median of C is a set of k points F of the universe U that minimizes ΣjεC {miniεF dij}.

Euclidean k-center 3 -center Minimize the maximum distance to the facility

Euclidean k-median 1 -median Minimize the average distance to the facility

Euclidean k-center • (Megiddo and Supowit, 1984): k-center is NP-hard. • (Feder and Green 1988): ε-approximation remains NPhard for any ε < (1+√ 7)/2 ≈ 1. 8229. Single facility location problem Megiddo (1983) proposed a linear time algorithm.

Euclidean k-median • Meggido and Supowit (1984): Problem is NP-hard in the plane, and cannot be approximated to within 3/2. Varignon frame (Classical Min. Sum)

Euclidean k-median • Meggido and Supowit (1984): Problem is NP-hard in the plane, and cannot be approximated to within 3/2. • Bajaj (1986) showed that the problem cannot be solved exactly in the plane using radicals. • Numerical methods are efficient. • Approximation scheme using ellipsoid method proposed by Chandrasekhar and Tamir (1990)

Settings Facility (service center) Collection Depots Client (demand service)

Application (1) Express Transportation

Application (2) Garbage collection

Survey of collection depots problem • Drezner and Weslowsky (2001) - on the line, an optimal Min. Sum solution must exist either on a customer point or a collection depot.

Survey of collection depots problem • Drezner and Weslowsky (2001) - on the line, an optimal Min. Sum solution must exist either on a customer point or a collection depot. - on a line, an optimal solution utilizes at most two depots.

Survey of collection depots problem Tamir and Halman (2005) studied the multi-facility Min. Max and Min. Sum depots problems in the plane, in graphs, in trees and in paths. - introduced three versions of the depots problem: • each customer is given a set of depots that the customer is allowed to use. • customer one-way: facility depot customer • depot one-way: facility customer depot

Survey of collection depots problem Tamir and Halman (2005): - 3 -approximation solution for the k-depots Min. Max problem in general graphs. - O(p+n 2 lgn) optimal solution for the k-depots Min. Max problem in trees. - For 1 -depot Min. Max problem • paths: • trees: • plane: O(n+p) O(p+n lgn) O(p 2 n 2 lg 3(pn)) (Euclidean) O(pn lg 4(pn) (Manhattan)

Survey of collection depots problem Benkoczi, Bhattacharya and Tamir (2007) considerd Min. Max and Min. Sum depots problems in trees. - 1 -depot Min. Sum : O((n+p)lg(n+p)) - 1 -depot Min. Max : O(n+p) - k-depot Min. Sum : O(p+kn 3)

Survey of collection depots problem Benkoczi, Bhattacharya, Das, and Sember (2005, 2008) considerd Min. Max and Min. Sum depots problems in the plane.

Outline of the rest of the talk • Our results (Min. Max and Min. Sum problems in the plane) -Voronoi diagrams -Feasible assignments -Approximation algorithms -The Barrier problem -The Room problem

Euclidean collection depots problem

Voronoi diagrams

Voronoi diagrams (Edges)

Voronoi diagrams (Properties)

Voronoi diagrams (Constructions)

Feasible assignments

Merged Voronoi diagrams of customers

Min. Sum depots problem

Min. Sum depots problem *

Min. Max depots problem • Min. Max depots problem can be solved in O(p 2 n 2) times the time it takes to solve Classical Min. Max for O(n) points. • Classical Min. Max problem can be solved in O(n) time (Megiddo(1983)) • Feasibility test of the Min. Max depots problem can be solved in O(n 2 p 2 lg(pn)) time (Tamir and Halman(2005)). This results in O(n 2 p 2 lg 3(pn)) solution for the general problem.

Barrier problem

Room problem

Classical Min. Max problem

Min. Max single collection depot problem