Submodular Percolation

Let $f:{\cal L}\to\mathbb{R}$ be a submodular function on a modular lattice ${\cal L}$; we show that there is a maximal chain ${\cal C}$ in ${\cal L}$ on which the sequence of values of $f$ is minimal among all paths from 0 to 1 in the Hasse diagram of ${\cal L}$, in a certain well-behaved partial order on sequences of reals. One consequence is that the maximum value of $f$ on ${\cal C}$ is minimized over all such paths—i.e., if one can percolate from 0 to 1 on lattice points $X$ with $f(X)\le b$, then one can do so along a maximal chain. The partial order on real sequences is defined by putting $\langle a(0),a(1),\dots,a(m)\rangle\preceq\langle b(0),\dots,b(n)\rangle$ 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 $a(i)\le b(j)$. Putting ${\bf a}\equiv{\bf b}$ if ${\bf a}\preceq{\bf b}\preceq{\bf a}$, 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.