  We study the ground state phase diagram of the quantum spin-$1/2$ Heisenberg
model on the kagom\'{e} lattice with first- ($J_1 < 0$), second- ($J_2 < 0$),
and third-neighbor interactions ($J_d > 0$) by means of analytical low-energy
field theory and numerical density-matrix renormalization group (DMRG) studies.
The results offer a consistent picture of the $J_d$-dominant regime in terms of
three sets of spin chains weakly coupled by the ferromagnetic inter-chain
interactions $J_{1,2}$. When either $J_1$ or $J_2$ is dominant, the model is
found to support one of two cuboctohedral phases, cuboc1 and cuboc2. These
cuboc states host non-coplanar long-ranged magnetic order and possess finite
scalar spin chirality. However, in the compensated regime $J_1 \simeq J_2$, a
valence bond crystal phase emerges between the two cuboc phases. We find
excellent agreement between an analytical theory based on coupled spin chains
and unbiased DMRG calculations, including at a very detailed level of
comparison of the structure of the valence bond crystal state. To our
knowledge, this is the first such comprehensive understanding of a highly
frustrated two-dimensional (2d) quantum antiferromagnet. We find no evidence of
either the one-dimensional (1d) gapless spin liquid or the chiral spin liquids,
which were previously suggested by parton mean field theories.
