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research
Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs
Authors
Yuichi Yoshida
Publication date
19 July 2010
Publisher
View
on
arXiv
Abstract
In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate
P
:
0
,
1
k
→
0
,
1
P:{0,1}^{k} \to {0,1}
P
:
0
,
1
k
→
0
,
1
except \equ where
k
≥
3
k\geq 3
k
≥
3
, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances
(
∣
P
−
1
(
0
)
∣
/
2
k
−
ϵ
)
(|P^{-1}(0)|/2^k-\epsilon)
(
∣
P
−
1
(
0
)
∣/
2
k
−
ϵ
)
-far from satisfiability requires
Ω
(
n
1
/
2
+
δ
)
\Omega(n^{1/2+\delta})
Ω
(
n
1/2
+
δ
)
queries where
n
n
n
is the number of variables and
δ
>
0
\delta>0
δ
>
0
is a constant that depends on
P
P
P
and
ϵ
\epsilon
ϵ
. This breaks a natural lower bound
Ω
(
n
1
/
2
)
\Omega(n^{1/2})
Ω
(
n
1/2
)
, which is obtained by the birthday paradox. We also show that every one-sided error tester requires
Ω
(
n
)
\Omega(n)
Ω
(
n
)
queries for such
P
P
P
. These results are hereditary in the sense that the same results hold for any predicate
Q
Q
Q
such that
P
−
1
(
1
)
⊆
Q
−
1
(
1
)
P^{-1}(1) \subseteq Q^{-1}(1)
P
−
1
(
1
)
⊆
Q
−
1
(
1
)
. For EQU, we give a one-sided error tester whose query complexity is
O
~
(
n
1
/
2
)
\tilde{O}(n^{1/2})
O
~
(
n
1/2
)
. Also, for 2-XOR (or, equivalently E2LIN2), we show an
Ω
(
n
1
/
2
+
δ
)
\Omega(n^{1/2+\delta})
Ω
(
n
1/2
+
δ
)
lower bound for distinguishing instances between
ϵ
\epsilon
ϵ
-close to and
(
1
/
2
−
ϵ
)
(1/2-\epsilon)
(
1/2
−
ϵ
)
-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances
(
1
−
2
k
/
2
k
−
ϵ
)
(1-2k/2^k-\epsilon)
(
1
−
2
k
/
2
k
−
ϵ
)
-far from satisfiability requires
Ω
(
n
)
\Omega(n)
Ω
(
n
)
queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the
d
d
d
-to-
1
1
1
Conjecture. As a corollary, for Maximum Independent Set on graphs with
n
n
n
vertices and a degree bound
d
d
d
, we show that every approximation algorithm within a factor
d/\poly\log d
and an additive error of
ϵ
n
\epsilon n
ϵ
n
requires
Ω
(
n
)
\Omega(n)
Ω
(
n
)
queries. Previously, only super-constant lower bounds were known
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Last time updated on 30/10/2017