1. 1 Maximum matching in graphs with an excluded minor Raphael Yuster University of Haifa Uri Zwick Tel Aviv University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

2. 2 Maximum matching algorithms Edmonds (1965) in P Micali-Vazirani (1980) (1994) O(mn 1/2 ) Lovász (1979) (randomized, cardinality) O(n ω ) Mucha-Sankowski (2004) (randomized) O(n ω ) Harvey (2006) (randomized) O(n ω ) As ω < 2.38 (Coppersmith-Winograd (1990)), the O(n ω )-time algorithm is faster on dense graphs.

3. 3 Planar graphs Micali-Vazirani (1980) (1994) O(n 1.5 ) Mucha-Sankowski (2004) (rand.) O(n ω/2 ) < O(n 1.19 ) Can the MS algorithm be extended to work for bounded genus graphs? Can the MS algorithm be extended to work for H-minor free graphs?

4. 4 Genus of surfaces The genus of a surface is R 3 is the is the largest number of non-intersecting simple closed curves that can be drawn on the surface without separating it. The sphere has genus 0. The torus has genus 1.

5. 5 Genus of graphs The genus of a graph is the smallest integer g such that the graph can be embedded on an (orientable) surface of genus g, without edge crossings. Planar graphs have genus 0. Genus-1 graphs are graphs that can be embedded on a torus

6. 6 K 5 has genus 1

7. 7 Graphs minors H is a minor of a G if H can be obtained from G by deleting and contracting edges G is H-minor free if H is not a minor of G

8. 8 Minor-free graphs Kuratowski (1930) Wagner (1937) K 3 -minor free K 4 -minor free forest series-parallel planar K 5 -minor free ∩ K 3,3 -minor free

9. 9 The graph minor theorem Finite graphs are well-quasi-ordered by the minor relation [Robertson-Seymour, (2004)] Equivalently, any infinite collection of finite graphs contains a graph which is a minor of another. Corollary: Every minor-closed family of graphs is characterized by a finite set of forbidden minors.

10. 10 New results Maximum matchings in bounded-genus graphs can be found in O(n ω/2 ) < O(n 1.19 ) time (rand.) Maximum matching in H-minor free graphs can be found in O(n 3ω/(3+ω) ) < O(n 1.326 ) time (rand.) The number of maximum matchings in bounded-genus graphs can be computed deterministically in O(n ω/2+1 ) < O(n 2.19 ) time

11. 11 Tutte (1947) From matchings to symbolic determinants Lovasz (1979) Determinants over Z p via randomizaition Lipton-Tarjan (1979) Planar separators Lipton-Rose-Tarjan (1979) Gilbert-Tarjan (1987) Nested dissection Rabin-Vazirani (1989) “Allowable” edges via matrix inversion Mucha-Sankowski (2006) Finding maximum matchings Maximum matchings in planar graphs

12. 12 Tutte’s matrix (Skew-symmetric symbolic adjacency matrix) 1 3 2 4 6 5

13. 13 Tutte’s theorem Let G=(V,E) be a graph and let A be its Tutte matrix. Then, G has a perfect matching iff det A  0. 1 3 2 4

14. 14 Tutte’s theorem Let G=(V,E) be a graph and let A be its Tutte matrix. Then, G has a perfect matching iff det A  0. Lovasz’s theorem Let G=(V,E) be a graph and let A be its Tutte matrix. Then, the rank of A is twice the size of a maximum matching in G.

15. 15 Separators A partition A,B,C of the vertices of G (k,α)-separates G iff A C B No edges connect A and B

16. 16 Separator tree A C B V BA

17. 17 Finding separators Lipton-Tarjan (1979): Planar graphs have (O(n 1/2 ), 2/3)-separators. Can be found in linear time. Alon-Seymour-Thomas (1990): H-minor free graphs have (O(n 1/2 ), 2/3)-separators. Can be found in O(n 1.5 ) time.  Reed and Wood (2005): For any ν>0, there is an O(n 1+ν )-time algorithm that finds (O(n (2  ν)/3 ), 2/3)-separators of H-minor free graphs.

18. 18 Nested dissection Lipton-Rose-Tarjan (1979) Gilbert-Tarjan (1987) Guassian elimination on a symmetric matrix with an (O(n β ),2/3)-separator tree can be performed in O(n β  ) time. Main idea: Let A,B,C be the first separator. Reorder the rows and columns in the order A,B,C. Reorder the rows of A and B recursively. Apply “lazy” Gaussian elimination on the reordered matrix using fast matrix multiplication.

19. 19 Running time of new algorithm

20. 20 Complications … Nested dissection works only if no 0’s are encountered on the diagonal (no pivoting). To ensure that, work with AA T instead of A. This corresponds to the squaring G. Unfortunately, G 2 is no longer planar (or H-minor free). Split the vertices of G to obtain a graph G’ of bounded degree. Thickened separators of G’ are also separators of (G’) 2.

21. 21 … … Vertex splitting Number of vertices unmatched by a maximum matching unchanged. But, splitting does not preserve H-minor freeness! Splitting can be done in a way that preserved planarity or bounded genus.

22. 22 Splitting introduces a K 4 -minor

23. 23 Main technical lemma Suppose that (O(n β ),2/3)-separators of H-minor free graphs can be found in O(n γ )-time. If G is an H-minor free graph, then a vertex-split version G’ of G of bounded degree and an (O(n β ),2/3)-separator tree of G’ can be found in O(n γ ) time.

24. 24 Open problems An O(n  /2 )-time algorithm for finding a maximum matching in H-minor free graphs? Faster algorithms for finding separators of H-minor free graphs? Faster combinatorial algorithms for finding maximum matchings in planar graphs?