On the cut-query complexity of approximating max-cut
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We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [RSW18]. This model arises as a natural special case of submodular function maximization: on query S ⊆ V, the oracle returns the total weight of the cut between S and V \ S. For most constants c ∈ (0,1], we nail down the query complexity of achieving a c-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at c = 1/2: we design a deterministic algorithm for global c-approximate max-cut in O(log n) queries for any c < 1/2, and show that any randomized algorithm requires Ω̃(n) queries to find a c-approximate max-cut for any c > 1/2. Additionally, we show that any deterministic algorithm requires Ω(n^2) queries to find an exact max-cut (enough to learn the entire graph), and develop a Õ(n)-query randomized c-approximation for any c < 1. Our approach provides two technical contributions that may be of independent interest. One is a query-efficient sparsifier for undirected weighted graphs (prior work of [RSW18] holds only for unweighted graphs). Another is an extension of the cut dimension to rule out approximation (prior work of [GPRW20] introducing the cut dimension only rules out exact solutions).
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