We consider the graph-theoretic problem of removing (few) nodes from a directed acyclic graph in order to reduce its depth. While this problem is intractable in the general case, we provide a variety of algorithms in the case where the graph is that of a circuit of fan-in (at most) two, and explore applications of these algorithms to secure multiparty computation with low communication. Over the past few years, a paradigm for low-communication secure multiparty computation has found success based on decomposing a circuit into low-depth "chunks". This approach was however previously limited to circuits with a "layered" structure. Our graphtheoretic approach extends this paradigm to all circuits. In particular, we obtain the following contributions:

-Fractionally linear-communication MPC in the correlated randomness model.

We provide an N -party protocol for computing an n-input, m-output F -arithmetic circuit with s internal gates (over any basis of binary gates) with communication complexity

at the cost of increasing the computational overhead from a small constant factor to a large one). Previously, comparable protocols either used more than s • N • log |F | bits of communication, required super-polynomial computation, were restricted to layered circuits, or tolerated a sub-optimal corruption threshold.

-Sublinear-Communication MPC. Assuming the existence of N -party Homomorphic Secret Sharing for logarithmic depth circuits (respectively doubly logarithmic depth circuits), we show there exists sublinear-communication secure N -party computation for all log 1+o(1) -depth (resp. (log log) 1+o(1) -depth) circuits. Previously, this result was limited to (O(log))-depth (resp. (O(log log))-depth) circuits, or to circuits with a specific structure (e.g. layered).

-The N 1 -OT complexity of MPC. We introduce the " N 1 -OT complexity of MPC " of a function f , denoted CN (f ), as the number of oracle calls required to securely compute f in the N 1 -OT hybrid model. We establish the following upper bound: for every

5 , where g(N ) is an explicit vanishing function.

We also obtain additional contributions to reducing the amount of bootstrapping for fully homomorphic encryption, and to other types of sublinear-communication MPC protocols such as those based on correlated symmetric private information retrieval.