Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance...
Transcript of Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance...
![Page 1: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/1.jpg)
Flip Distances between Graph OrientationsJean Cardinal
Joint work with Oswin Aichholzer, Tony Huynh, Kolja Knauer, Torsten Mütze, Raphael
Steiner, and Birgit Vogtenhuber
October 2020
1 / 21
![Page 2: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/2.jpg)
Flip Graphs
2 / 21
![Page 3: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/3.jpg)
Polytope
3 / 21
![Page 4: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/4.jpg)
Flip Distances
• Diameter of associahedraSleator, Tarjan, Thurston 1988, Pournin 2014
• Computational question: given two objects, what is their flipdistance?• Flip distance between triangulations of point sets is NP-hard.
Lubiw and Pathak 2015
• Flip distance between triangulations of simple polygons isNP-hard.
Aichholzer, Mulzer, Pilz 2015
• Flip distance between triangulations of a convex polygon:Major open question.
4 / 21
![Page 5: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/5.jpg)
Flip Distances
Complexity of flip distances on “nice” combinatorial polytopes?• Matroid polytopes: easy• Associahedra and polymatroids: open• Intersection of two matroids?
5 / 21
![Page 6: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/6.jpg)
α-orientations
An α-orientation of G is an orientation of the edges of G in whichevery vertex v has outdegree α(v).
6 / 21
![Page 7: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/7.jpg)
Perfect Matchings in Bipartite Graphs
1
1
1
1
1
1
2 2
12
7 / 21
![Page 8: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/8.jpg)
Flips in α-orientations
1
1
1
1
1
1
2 2
12
1
1
1
1
1
1
2 2
12
8 / 21
![Page 9: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/9.jpg)
Flip Graph on Perfect Matchings
9 / 21
![Page 10: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/10.jpg)
Adjacency
• The common base polytope of two matroids is the intersectionof the two matroid polytopes.• α-orientations are intersections of two partition matroids.• Adjacency is characterized by cycle exchanges.
Frank and Tardos 1988, Iwata 2002
• Adjacency on perfect matching polytope: symmetric di�erenceinduces a single cycle.
Balinski and Russako� 1974
10 / 21
![Page 11: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/11.jpg)
Adjacency
A = {1, 3, 5, 7, 8}2
46
1 3
5
7 8
B = {2, 4, 6, 7, 8}
1 3
5
7 8
6 4
2
5
3
1
6
4
2A \ B B \ A
11 / 21
![Page 12: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/12.jpg)
Flip Distance
2 1
1
1
2 1
1
1
2 1
1
1
2 1
1
1
2 1
1
1
C1 C2 C3 C4
D1
D2
D3
• Symmetric di�erence composed of four cycles: C1, C2, C3, C4
• Flip distance three: D1, D2, D3
12 / 21
![Page 13: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/13.jpg)
NP-completeness
TheoremGiven a two-connected bipartite subcubic planar graph G and a pairX ,Y of perfect matchings in G, deciding whether the flip distancebetween X and Y is at most two is NP-complete.
13 / 21
![Page 14: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/14.jpg)
NP-hardness
Reduction from Hamiltonian cycle in planar, degree 1/2-2/1digraphs.
14 / 21
![Page 15: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/15.jpg)
Planar Duality
15 / 21
![Page 16: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/16.jpg)
Dual Flips
A c-orientation is an orientation in which the number of forwardedges in any cycle C is equal to c(C).
Propp 2002, Knauer 2008
primal dualα-orientation c-orientation
directed cycle flip directed cut flipfacial cycle flip source/sink flip
16 / 21
![Page 17: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/17.jpg)
Source/sink flips
• What is the complexity of computing the source/sink flipdistance?
• Here the flip graphs is known to have a nice structure.Propp 2002, Felsner and Knauer 2009,2011
17 / 21
![Page 18: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/18.jpg)
A Distributive La�ice
18 / 21
![Page 19: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/19.jpg)
Source/sink flip distance
TheoremThere is an algorithm that, given a graph G with a fixed vertex > and apair X ,Y of c-orientations of G, outputs a shortest source/sink flipsequence between X and Y, and runs in time O(m3) where m is thenumber of edges of G.
19 / 21
![Page 20: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/20.jpg)
Flip distance with Larger Cut Sets
TheoremLet X ,Y be c-orientations of a connected graph G with fixed vertex >.It is NP-hard to determine the length of a shortest cut flip sequencetransforming X into Y , which consists only of minimal directed cutswith interiors of order at most two.
Reduction from the problem of computing the jump number of aposet.
20 / 21
![Page 21: Flip Distances between Graph Orientationspak/seminars/Slides/Cardinal.pdfSource/sink flip distance Theorem There is an algorithm that, given a graph G with a fixed vertex >and a pair](https://reader035.fdocuments.net/reader035/viewer/2022071505/612559ce662b2b4b5d1148be/html5/thumbnails/21.jpg)
Thank you!
Algorithmicah�ps://doi.org/10.1007/s00453-020-00751-1
21 / 21