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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