Hardness and algorithms for variants of line graphs of ...beaudou/docs/isaac_2013.pdfHardness and algorithms for variants of line graphs of directed graphs ... In Tohoku Mathematical

Hardness and algorithms for variants of

line graphs of directed graphs

Mourad Baïou, Laurent Beaudou, Zhentao Li and VincentLimouzy

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

ISAAC 2013, Hong-Kong, December 16th, 2013

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∑

dℓ1i

median : min∑

dℓ2i

median : min∑

dℓ2i

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 andthe 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

min∑

cijxij +∑

ciyi

xij ≤ yj

yi +∑

i→j xij = 1

Facility location graph

Facility location graph

Facility location graph

Independent Set

Smelling like line graphs

✘ · · Line graphs of bipartite graphs✔ ✔ ✔ Line graphs✔ ✘ ✘ Chvátal and Ebenegger✔ ✔ ✘ Our case

Do you recognize me ?

Given a digraph D,

underlying graph of the line digraph of Dnp-complete

⊓

flg(D)⊓

line graph of the underlying graph of Dp

Do you recognize me ?

Given a digraph D,

underlying graph of the line digraph of Dnp-complete

⊓

flg(D)⊓

line graph of the underlying graph of Dp

Theorem [Baïou, B., Li and Limouzy, 2013+]

Recognizing if a graph G is a facility location graph isnp-complete.

Did you see my big gadget ?

Triangle-free graphs

Lemma

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

G G ′

Triangle-free graphs

Theorem [BBLL, 2013+]

If G is triangle-free, then G is a facility location graph if andonly if, once peeled off, every connected component has atmost one cycle.

Triangle-free graphs

Theorem [BBLL, 2013+]

If G is triangle-free, then G is a facility location graph if andonly if, once peeled off, every connected component has atmost one cycle.

This yields an infinite family of forbidden induced subgraphs.

Triangle-free graphs

Triangle-free graphs

FLG is not minor closed

The slide where Erdős is mentioned

Corollary

Triangle-free facility location graphs are 3-colourable.

The slide where Erdős is mentioned

Paul Erdős

Budapest 1913 - Warsaw 1996

Theorem

There exist graphs with arbitrarily highgirth and chromatic number.

Canad. J. Math., 1959

Jan Mycielski

Wisniowa 1932

Theorem

I can construct such graphs for girth 4and any chromatic number.

Colloquium Math., 1955

Colouring continued

Theorem [BBLL, 2013+]

The vertex colouring of facility location graphs isnp-complete.

Reduction from chromatic index of simple graphs (thanks toIan Holyer’s theorem)

Stable set on facility location graphs

Theorem [Poljak, 1974]

The maximum stable set problem is np-complete intriangle-free facility location graphs.

The 3-coloring gives a 3-approx.

Stable set on facility location graphs

Corollary

UFLP is np-complete, even if some graphs are forbidden.

What’s next ?

What’s next ?

What’s next ?

No simple algorithm to detect suchcycles

What’s next ?

A (small) set of obstructions that make UFLP polynomial

What’s next ?

Jim Morrison

Melbourne, Florida 1943 - Paris 1971

« This is the end. »The Doors, 1967