Improved Lower Bounds for Submodular Function Minimization
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We provide a generic technique for constructing families of submodular functions to obtain lower bounds for submodular function minimization (SFM). Applying this technique, we prove that any deterministic SFM algorithm on a ground set of n elements requires at least Ω(n log n) queries to an evaluation oracle. This is the first super-linear query complexity lower bound for SFM and improves upon the previous best lower bound of 2n given by [Graur et al., ITCS 2020]. Using our construction, we also prove that any (possibly randomized) parallel SFM algorithm, which can make up to 𝗉𝗈𝗅𝗒(n) queries per round, requires at least Ω(n / log n) rounds to minimize a submodular function. This improves upon the previous best lower bound of Ω̃(n^1/3) rounds due to [Chakrabarty et al., FOCS 2021], and settles the parallel complexity of query-efficient SFM up to logarithmic factors due to a recent advance in [Jiang, SODA 2021].
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