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research
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Authors
Ryan O'Donnell
John Wright
Chenggang Wu
Yuan Zhou
Publication date
10 January 2014
Publisher
Doi
Cite
View
on
arXiv
Abstract
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic
n
n
n
-vertex graphs
G
G
G
and
H
H
H
such that any sum-of-squares (SOS) proof of nonisomorphism requires degree
Ω
(
n
)
\Omega(n)
Ω
(
n
)
. In other words, we show an
Ω
(
n
)
\Omega(n)
Ω
(
n
)
-round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs
G
G
G
and
H
H
H
which are not even
(
1
−
1
0
−
14
)
(1-10^{-14})
(
1
−
1
0
−
14
)
-isomorphic. (Here we say that two
n
n
n
-vertex,
m
m
m
-edge graphs
G
G
G
and
H
H
H
are
α
\alpha
α
-isomorphic if there is a bijection between their vertices which preserves at least
α
m
\alpha m
α
m
edges.) Our second result is that under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust} Graph Isomorphism problem is hard. I.e.\ for every
ϵ
>
0
\epsilon > 0
ϵ
>
0
, there is no efficient algorithm which can distinguish graph pairs which are
(
1
−
ϵ
)
(1-\epsilon)
(
1
−
ϵ
)
-isomorphic from pairs which are not even
(
1
−
ϵ
0
)
(1-\epsilon_0)
(
1
−
ϵ
0
​
)
-isomorphic for some universal constant
ϵ
0
\epsilon_0
ϵ
0
​
. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest
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info:doi/10.1137%2F1.978161197...
Last time updated on 01/04/2019