A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization

The problem of minimizing a submodular function (SFM) is a common generalization of several fundamental combinatorial optimization problems, including minimum
$s-t$
cuts in graphs and matroid intersection. It is well-known that a submodular function can be minimized with only
$\text{poly} (N)$
function evaluation queries where
$N$
denotes the universe size. However, all known polynomial query algorithms for SFM are highly adaptive, requiring at least
$N$
rounds of adaptivity. A natural question is if SFM can be efficiently solved in a highly parallel manner, namely, with
$\text{poly} (N)$
queries using only poly-logarithmic rounds of adaptivity. An important step towards understanding the adaptivity needed to solve SFM efficiently was taken in the very recent work of Balkanski and Singer who showed that any SFM algorithm with
$\text{poly} (N)$
queries. This left open the possibility of efficient SFM algorithms with poly-logarithmic rounds of adaptivity. In this work, we strongly rule out this possibility by showing that any, possibly randomized, algorithm for submodular function minimization making
$\text{poly} (N)$
queries requires
$\tilde{\Omega}(N^{1/3})$
rounds of adaptivity. In fact, we show a polynomial lower bound on the number of rounds of adaptivity even for algorithms that make up to
$2^{N^{1-\delta}}$
queries, for any constant
$\delta > 0$
.