On a facility location problem

Mourad Baïou, Laurent Beaudou and Vincent Limouzy

LIMOS, Université Blaise-Pascal, Clermont-Ferrand, France

BGW ’12, Bordeaux, November 21st, 2012

Cast (in order of appearance)

Pierre de Fermat

Beaumont-de-L. 1601 - Castres 1665

Methodus de maxima et minima, 1638

Cast (in order of appearance)

Evangelista Torricelli

Faenza 1608 - Florence 1647

V. Viviani, De maximis et minimis..., 1659

centroid : min∑

d2i

min∑

d ℓ1i

median : min∑

d ℓ2i

median : min∑

d ℓ2i

Cast (in order of appearance)

Pierre Varignon

Caen 1654 - Paris 1722

Cast (in order of appearance)

Alfred Weber

Erfurt 1868 - Heidelberg 1958

Über den Standort der Industrien, 1909

Cast (in order of appearance)

Endre Weiszfeld

Budapest 1916 - Santa Rosa 2003

In Tohoku Mathematical Journal 43, 1937

Cast (in order of appearance)

G. B. Dantzig W. M. Hirsch

1914 - 2005 1918 - 2007

S. L. Hakimi

The fixed charge problem

Naval Research Logistics Quarterly, 1968

Optimal location of switching centers and

the absolute centers and medians of a graph

Operations Research, 1964

The problem

The problem

The problem

cj

The problem

cj

cij

The problem

cj

cij

The problem

cj

cij

xij ≤ yj

The problem

cj

cij

xij ≤ yj

yi +∑

i→j xij = 1

The problem

cj

cij

max∑

cijxij −∑

ciyi

xij ≤ yj

yi +∑

i→j xij = 1

Facility location graph

Facility location graph

Facility location graph

Independent Set

Questions

What do these guys look like ?

What conditions imply a nice polytope for the digraph ?

What’s new ?

Lemma

G is a facility location graph if and only if G ′ is a facility

location graph.

G G ′

What’s new ?

Theorem

If G is triangle-free, then G is a facility location graph if and

only if, once peeled off, every connected component has at

most one cycle.

What’s new ?

Theorem

If G is triangle-free, then G is a facility location graph if and

only if, once peeled off, every connected component has at

most one cycle.

This yields an infinite family of forbidden induced subgraphs.

Sun1 Sun2

What’s new ?

What’s new ?

FLG is not minor closed

What’s new ?

Corollary

Triangle-free facility location graphs are 3-colourable.

What’s new ?

Corollary

Triangle-free facility location graphs are 3-colourable.

Refinement for directed graphs.

What’s next ?

What’s next ?

What’s next ?

No simple algorithm to detect such

cycles

What’s next ?

Graph theoretic questions :

1. Complexity of recognition.

2. Complexity of Independent Set in this class.

Linear programming questions :

1. How to transpose the results on Independent Set to

this facility location problem.

2. How to describe associated polytopes.

What’s next ?

Jim Morrison

Melbourne, Florida 1943 - Paris 1971

« This is the end. »

The Doors, 1967