Hardness and algorithms for variants of line graphs of ...beaudou/docs/isaac_2013.pdfHardness and...
Transcript of Hardness and algorithms for variants of line graphs of ...beaudou/docs/isaac_2013.pdfHardness and...
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
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
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.
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.
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.