  We derive several new reformulations of the Hubeny-Rangamani-Takayanagi
covariant holographic entanglement entropy formula. These include: (1) a
minimax formula, which involves finding a maximal-area achronal surface on a
timelike hypersurface homologous to D(A) (the boundary causal domain of the
region A whose entropy we are calculating) and minimizing over the
hypersurface; (2) a max V-flow formula, in which we maximize the flux through
D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm
that is non-local in time; and (3) a min U-flow formula, in which we minimize
the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject
to a lower bound on its norm that is non-local in space. The two flow formulas
define convex programs and are related to each other by Lagrange duality. For
each program, the optimal configurations dynamically find the HRT surface and
the entanglement wedges of A and its complement. The V-flow formula is the
covariant version of the Freedman-Headrick bit thread reformulation of the
Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a
"thread distribution", and explain how Riemannian flows, V-flows, and U-flows
can be expressed in terms of thread distributions.
