On the Simultaneous Minimum Spanning Trees Problem

@article{Konecn2018OnTS,
  title={On the Simultaneous Minimum Spanning Trees Problem},
  author={Matej Konecn{\'y} and Stanislav Kucera and Jana Novotn{\'a} and Jakub Pek{\'a}rek and Martin Smol{\'i}k and Jakub Tetek and Martin T{\"o}pfer},
  journal={ArXiv},
  year={2018},
  volume={abs/1712.00253}
}
Simultaneous Embedding with Fixed Edges (SEFE) is a problem where given $k$ planar graphs we ask whether they can be simultaneously embedded so that the embedding of each graph is planar and common edges are drawn the same. Problems of SEFE type have inspired questions of Simultaneous Geometrical Representations and further derivations. Based on this motivation we investigate the generalization of the simultaneous paradigm on the classical combinatorial problem of minimum spanning trees. Given… 

References

SHOWING 1-10 OF 13 REFERENCES
Efficient Algorithms for Graphic Matroid Intersection and Parity (Extended Abstract)
TLDR
Improved algorithms for other problems are obtained, including maintaining a minimum spanning tree on a planar graph subject to changing edge costs, and finding shortest pairs of disjoint paths in a network.
On the History of the Minimum Spanning Tree Problem
TLDR
There are several apparently independent sources and algorithmic solutions of the minimum spanning tree problem and their motivations, and they have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century.
On the shortest spanning subtree of a graph and the traveling salesman problem
7. A. Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull. Acad. Sei. URSS, Ser. Math. vol. 5 (1941) pp. 233-240. 8. J.
Simultaneous Embedding of Planar Graphs
TLDR
A survey of recent work investigating simultaneous embedding problems both from a theoretical and a practical point of view.
An optimal minimum spanning tree algorithm
TLDR
It is established that the algorithmic complexity of the minimumspanning tree problem is equal to its decision-tree complexity and a deterministic algorithm to find aminimum spanning tree of a graph with vertices and edges that runs in time is presented.
Submodular Functions, Matroids, and Certain Polyhedra
  • J. Edmonds
  • Mathematics
    Combinatorial Optimization
  • 2001
The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the
Reducibility Among Combinatorial Problems
  • R. Karp
  • Computer Science
    50 Years of Integer Programming
  • 1972
TLDR
Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.
Shortest connection networks and some generalizations
The basic problem considered is that of interconnecting a given set of terminals with a shortest possible network of direct links. Simple and practical procedures are given for solving this problem
, and Matthias Stallmann . ” Efficient algorithms for graphic matroid intersection and parity
  • International Colloquium on Automata , Languages , and Programming
  • 1985
” O jistém problému minimı́lńım ( About a certain minimal problem ) ” , Práce mor . př́ırodověd . spol
  • : ” O jistém problému minimálńım ” , Práce Moravské Př́ırodovědecké Společnosti
  • 1926
...
...