Expanders, Universal Graphs and Disjoint Paths

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Expanders, Universal Graphs and Disjoint Paths
Noga Alon Joint work with Michael Capalbo
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Eigenvalues and Expanders
Expanders are constant-degree graphs in which
every set X of at most half the vertices has
O(X) neighbors outside X.
For regular graphs (with a loop in each
vertex) this is equivalent to a spectral property
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Tanner, A-Milman, A A regular graph is a good
AChung The number of edges in any set X
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Disjoint Paths
The Problem Given r pairs of distinct vertices
s1t1 ,s2t2 srtr in an expander G on n
vertices, find edge-disjoint paths P1, P2,Pr
where Pi is a path from si to ti
The larger r is, the harder it is to find the
paths
Motivation Communication networks, Distributed
memory Computer architecture
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History
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The new result
Moreover -Each vertex can appear in up to d/3
pairs
-The algorithm is online
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s1
t2
s2
t1
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The result-more precisely
Definition A d-regular graph G on n vertices is
a very strong expander if -the average degree
in any subgraph with at most n/10 vertices is at
most d/6 -the average degree in any subgraph with
at most n/2 vertices is at most 2d/3
The adversary-Router game on G in each round
i the adversary picks a pair of vertices siti,
and the router has to find a path Pi connecting
them, keeping all paths edge disjoint.
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Theorem If G is a d-regular very strong
expander on n vertices, and
then the router can win any r-round
adversary- router game on G by a deterministic
polynomial time algorithm, assuming the adversary
does not choose any vertex as an endpoint si or
ti more than d/3 times.
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A brief outline of the proof (for disjoint pairs)
During the game, call a vertex not yet picked by
the adversary available, and an edge not yet
used by the router remaining.
The router maintains in round i -A subgraph Hi
of G, consisting of remaining edges, in which
all degrees are at least 3d/4 2 -A set of edge
disjoint paths Qs consisting of remaining edges,
from each available vertex s to Hi.
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Two crucial facts -A subgraph of minimum degree
3d/4 in a very strong d-regular expander is
itself a pretty good expander
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-If G is a very strong d-regular expander on n
vertices, Q is a set of edges of G and one keeps
removing vertices of G until the minimum degree
is at least 3d/4 2, then the remaining graph
has at least n-15Q/d vertices.
Indeed, otherwise the set of deleted
vertices will span too many edges.
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-In each round, the router finds a short path in
Hi that augments the paths from si and ti to Hi
and provide the path Pi. -She deletes the edges
of Pi, updates Hi, and finds new paths Qs using
a Network Flow algorithm.
si
ti
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Open
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Universal Graphs Definition H- A family of
graphs. G is H- universal if it contains every
member of H as a subgraph
Example G
is H-universal for the family of all 2-regular
graphs on 7 vertices
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Objective Construct sparse H-universal
graphs for interesting families H Motivation
VLSI circuit design
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Universal graphs for bounded-degree
graphs H(k,n)all graphs on n vertices,
max-degree k Question Estimate the minimum
possible number of edges of an H(k,n)-universal
graph. A,Capalbo,Kohayakawa,Rödl,Rucinski,Szemeré
di (00) O(n2-2/k) edges are needed, O(n2-1/k
log1/k n) suffice ACKRRS (01) O(n2-2/k log
18/k n) edges suffice
A, Capalbo (06) O(n2-2/k log 4/k n) edges
suffice
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New Theorem For all k 3 there is cc(k) and
an explicit H(k,n)-universal G with at most c
n2-2/k edges. The proof applies properties of
high-girth expanders and provides a
deterministic embedding procedure.
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The proof for even k is simpler, the one for odd
k requires an additional effort a new graph
decomposition result.
Theorem for k4 The minimum possible number of
edges of a graph that contains a copy of every
graph on n vertices with maximum degree at most
4 is T(n3/2)
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The lower bound Simple counting there are
many 4-regular graphs on n vertices, and a
graph with m edges cannot contain too many
subgraphs with 2n edges The upper
bound Construction using high-girth expanders
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The construction Let a,d be absolute constants,
put ma n1/2, and let F be a d-regular Ramanujan
expander of girth at least ? logd-1m. Thus all
nontrivial eigenvalues of F are of absolute value
at most 2(d-1)1/2. Define G(V,E), where
V(V(F))2 and (a1,a2) is adjacent to (b1,b2) iff
ai and bi are within distance 2 in F for i1
and/or i2. Clearly EO(n3/2). Main claim G
is H(4,n)-universal.
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A homomorphism from a graph Z to a graph T is
a mapping of V(Z) to V(T) such that adjacent
vertices in Z are mapped to adjacent vertices in
T. Thus there is an injective homomorphism from
Z to T iff Z is a subgraph of T. Pn - the path
of length n. A homomorphism from Pn to F is a
walk on F. The k-th power Tk of a graph T is the
graph on V(T) in which two vertices are adjacent
iff their distance in T is at most k.
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Let H be a graph on n vertices with maximum
degree at most 4. By Petersens Theorem H can be
decomposed into two spanning subgraphs H1,H2,
each having max. degree at most 2. There are
bijective homomorphisms gi from Hi to Pn2 To
embed H in G we define homomorphisms fi from
Pn to F so that f(v)(f1(g1(v)), f2(g2(v)) ) is
an injective homomorphism from H to G.This is
done by defining each fi as an appropriate
non-back-tracking walk on F.
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The properties of F A (simple) walk of length q
in F is a sequence Ww0,w1, ,wq of distinct
vertices of F, with wiwi1 being an edge for all
i. If S1, S2, ,Sq are subsets of V(F), then W
slips by the sets Si if for all i, wi is not in
Si
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A vertex w is nice with respect to the sets Si
if there are at least m/2 vertices z so that
there is a walk ww0,w1, ,wqz of length q
that starts at w, slips by the sets Si,and ends
at z. A vertex w is very nice with respect to
the sets Si if for every set of vertices Q
containing at most d/20 log d neighbors of
each vertex, w is nice with respect to the sets
Si Q.
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Lemma Let F be a high-girth Ramanujan expander
on m vertices (as above), and put qlog m/ log
10. Then, for any collection of sets S1,S2, Sq
of vertices of F satisfying Si m/20 for all
i, the number of vertices w that are very nice
with respect to the sets Si is at least
9m/10. The proof uses the spectral properties
of F and the fact it has high girth.
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The homomorphism f1 from Pn to F can be any
walk in F covering no vertex more than n1/2
times. Given f1, one can define f2
deterministically in steps, where is each step it
is defined on q consecutive vertices of the
path, making sure that the mapping
f(v)(f1(g1(v)), f2(g2(v))) is injective. To do
so, the walk has to slip by appropriately
defined sets. The difficulty is that these sets
change during the process. The notions nice
and very nice help to overcome this difficulty.
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A crucial observation When augmenting the sets
Si by the vertices of a walk of length q, every
vertex v which was very nice (with respect to the
sets Si), becomes nice (with respect to the
augmented sets).
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The construction of universal graphs for H(k,n)
with kgt4 even is similar The odd case requires
more efforts
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A graph is thin if every connected component of
it is either a subgraph of a cycle with pendant
edges
or a graph with max. degree 3 and at most two
vertices of degree 3
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Fact Every thin graph can be mapped
homomorphically and bijectively to the forth
power of a path. Theorem Let H be a graph of
maximum degree k. Then there are k thin spanning
subgraphs H1, H2, ,Hk of H, so that each edge
of H lies in two of the graphs Hi.
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A universal graph for H(k,n) Let F be a
high-girth Ramanujan graph on ma n1/k vertices.
Construct G(V(G),E(G)) as follows V(G)(V(F))k
(a1,a2, ,ak) and (b1,b2, ,bk) are adjacent
iff there are at least two indices i so that ai
and bi are within distance 4 in F.
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Open Is there an H(k,n)-universal graph on n
vertices with O(n22/k) edges ?