We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to long-standing decomposition problems. For instance, our results imply the following. Let
G
G
be a quasi-random
n
n
-vertex graph and suppose
H
1
,
…
,
H
s
H_1,\dots ,H_s
are bounded degree
n
n
-vertex graphs with
∑
i
=
1
s
e
(
H
i
)
≤
(
1
−
o
(
1
)
)
e
(
G
)
\sum _{i=1}^{s} e(H_i) \leq (1-o(1)) e(G)
. Then
H
1
,
…
,
H
s
H_1,\dots ,H_s
can be packed edge-disjointly into
G
G
. The case when
G
G
is the complete graph
K
n
K_n
implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach problem.
We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemerédi’s regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy, and Szemerédi to the setting of approximate decompositions.