The Submodular Orienteering problem asks for an $s$-$t$ walk in a directed graph that maximizes a monotone submodular function over the visited vertices subject to a bound on its length. We give a deterministic polynomial-time approximation algorithm with subpolynomial approximation rate $2^{O(\sqrt{\log n\,\log\log n})}$. This improves over the $O(n^\varepsilon)$ approximation rate that was previously known for every fixed $\varepsilon>0$. We prove a decomposition lemma that partitions a directed graph into rooted subgraphs such that every path intersects only polylogarithmically many such subgraphs. Our algorithm uses this decomposition to construct its recursive subproblems.

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