We study quasi–Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of
ε
\varepsilon
depends on
ε
−
1
{\varepsilon }^{-1}
and the dimension
s
s
. Strong tractability means that it does not depend on
s
s
and is bounded by a polynomial in
ε
−
1
{\varepsilon }^{-1}
. The least possible value of the power of
ε
−
1
{\varepsilon }^{-1}
is called the
ε
\varepsilon
-exponent of strong tractability. Sloan and Woźniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the
ε
\varepsilon
-exponent of strong tractability is between 1 and 2. However, their proof is not constructive. In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with
ε
\varepsilon
-exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Woźniakowski’s assumption. We show that quasi–Monte Carlo algorithms using Niederreiter’s
(
t
,
s
)
(t,s)
-sequences and Sobol sequences achieve the optimal convergence order
O
(
N
−
1
+
δ
)
O(N^{-1+\delta })
for any
δ
>
0
\delta >0
independent of the dimension with a worst case deterministic guarantee (where
N
N
is the number of function evaluations). This implies that strong tractability with the best
ε
\varepsilon
-exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter’s
(
t
,
s
)
(t,s)
-sequences and Sobol sequences.