We show that the deterministic past history of the Universe can be uniquely
reconstructed from the knowledge of the present mass density field, the latter
being inferred from the 3D distribution of luminous matter, assumed to be
tracing the distribution of dark matter up to a known bias. Reconstruction
ceases to be unique below those scales -- a few Mpc -- where multi-streaming
becomes significant. Above 6 Mpc/h we propose and implement an effective
Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the
Zel'dovich approximation is well satisfied and reconstruction becomes an
instance of optimal mass transportation, a problem which goes back to Monge
(1781). After discretization into N point masses one obtains an assignment
problem that can be handled by effective algorithms with not more than cubic
time complexity in N and reasonable CPU time requirements. Testing against
N-body cosmological simulations gives over 60% of exactly reconstructed points.
We apply several interrelated tools from optimization theory that were not
used in cosmological reconstruction before, such as the Monge-Ampere equation,
its relation to the mass transportation problem, the Kantorovich duality and
the auction algorithm for optimal assignment. Self-contained discussion of
relevant notions and techniques is provided.Comment: 26 pages, 14 figures; accepted to MNRAS. Version 2: numerous minour
clarifications in the text, additional material on the history of the
Monge-Ampere equation, improved description of the auction algorithm, updated
bibliography. Version 3: several misprints correcte