  The paper starts with discussion of applications of cycle spaces to
transportation cost. After a short survey of the known results on cycle spaces,
we turn to the study of minimal projections onto cycle spaces in the
corresponding $\ell_1$-spaces. This study is naturally related to the study of
invariant projections on the cycle space, which, in turn, are determined by the
properties of representations of the automorphism group of the corresponding
graph. The main focus is on discrete tori and Hamming graphs.
