Submodular percolation

Let f : L -> R be a submodular function on a modular lattice L; we show that there is a maximal chain C in L on which the sequence of values of f is minimal among all paths from 0 to 1 in the Hasse diagram of L, in a certain well-behaved partial order on sequences of reals. One consequence is that the maximum value of f on C is minimized over all such paths-i.e., if one can percolate from 0 to 1 on lattice points X with f( X) <= b, then one can do so along a maximal chain. The partial order on real sequences is defined by putting < a(0), a(1), ... , a(m)> <= < b(0), ... , b(n)> if there is a way to "schedule" the sequences starting at (a(0), b(0)) and ending at (a(m), b(n)) so that always a(i) <= b(j). Putting a equivalent to b if a <= b <= a, each equivalence class has a unique shortest sequence which we call a worm. We use the properties of worms to give an efficient method to schedule many real sequences in parallel. The results in the paper are applied in a number of other settings, including obstacle navigation, graph search, coordinate percolation, and finding a lost child in a field.