  The approximate contraction of a Projected Entangled Pair States (PEPS)
tensor network is a fundamental ingredient of any PEPS algorithm, required for
the optimization of the tensors in ground state search or time evolution, as
well as for the evaluation of expectation values. An exact contraction is in
general impossible, and the choice of the approximating procedure determines
the efficiency and accuracy of the algorithm. We analyze different previous
proposals for this approximation, and show that they can be understood via the
form of their environment, i.e. the operator that results from contracting part
of the network. This provides physical insight into the limitation of various
approaches, and allows us to introduce a new strategy, based on the idea of
clusters, that unifies previous methods. The resulting contraction algorithm
interpolates naturally between the cheapest and most imprecise and the most
costly and most precise method. We benchmark the different algorithms with
finite PEPS, and show how the cluster strategy can be used for both the tensor
optimization and the calculation of expectation values. Additionally, we
discuss its applicability to the parallelization of PEPS and to infinite
systems (iPEPS).
