Maximum matching in graphs
with an excluded minor
Raphael Yuster
University of Haifa
Uri Zwick
Tel Aviv University
1
Maximum matching algorithms
Edmonds (1965)
in P
Micali-Vazirani (1980) (1994)
O(mn1/2)
Lovász (1979) (randomized, cardinality)
O(nω)
Mucha-Sankowski (2004) (randomized)
O(nω)
Harvey (2006) (randomized)
O(nω)
As ω < 2.38 (Coppersmith-Winograd (1990)),
the O(nω)-time algorithm is faster on dense graphs.
2
Planar graphs
Micali-Vazirani (1980) (1994)
Mucha-Sankowski (2004) (rand.)
O(n1.5)
O(nω/2) < O(n1.19)
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?
3
Genus of surfaces
The genus of a surface is R3 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.
4
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
5
K5 has genus 1
6
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
7
Minor-free graphs
K3-minor free
forest
K4-minor free
series-parallel
K5-minor free ∩ K3,3-minor free
planar
Kuratowski (1930)
Wagner (1937)
8
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.
9
New results
Maximum matchings in bounded-genus graphs
can be found in O(nω/2) < O(n1.19) time (rand.)
Maximum matching in H-minor free graphs can
be found in O(n3ω/(3+ω)) < O(n1.326) time (rand.)
The number of maximum matchings in
bounded-genus graphs can be computed
deterministically in O(nω/2+1) < O(n2.19) time
10
Maximum matchings in planar graphs
Tutte (1947)
From matchings to
symbolic determinants
Lovasz (1979)
Determinants over Zp
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
11
Tutte’s matrix
(Skew-symmetric symbolic adjacency matrix)
4
1
6
3
2
5
12
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
2
4
3
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.
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.
14
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
15
Separator tree
A
B
C
V
A
B
16
Finding separators
Lipton-Tarjan (1979):
Planar graphs have (O(n1/2), 2/3)-separators.
Can be found in linear time.
Alon-Seymour-Thomas (1990):
H-minor free graphs have (O(n1/2), 2/3)-separators.
Can be found in O(n1.5) time. 
Reed and Wood (2005):
For any ν>0, there is an O(n1+ν)-time algorithm
that finds (O(n(2ν)/3) , 2/3)-separators
of H-minor free graphs.
17
Nested dissection
Lipton-Rose-Tarjan (1979) Gilbert-Tarjan (1987)
Guassian elimination on a symmetric matrix with
an (O(nβ),2/3)-separator tree can be performed in
O(nβ) 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.
18
Running time of new algorithm
n
1+ º
+n
2¡ º
3
Choose º =
n
!
2! ¡ 3
3+ !
3!
3+ !
19
Complications …
Nested dissection works only if no 0’s are
encountered on the diagonal (no pivoting).
To ensure that, work with AAT instead of A.
This corresponds to the squaring G.
Unfortunately, G2 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.
20
Vertex splitting
…
…
Number of vertices unmatched
by a maximum matching unchanged.
Splitting can be done in a way that preserved
planarity or bounded genus.
But, splitting does not preserve
H-minor freeness!
21
Splitting introduces a K4-minor
22
Main technical lemma
Suppose that (O(nβ),2/3)-separators of
H-minor free graphs can be found in
O(nγ)-time.
If G is an H-minor free graph, then a
vertex-split version G’ of G of bounded
degree and an (O(nβ),2/3)-separator tree
of G’ can be found in O(nγ) time.
23
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?
24