On Multidimensional and Monotone k-SUM

Abstract

The well-known k-SUM conjecture is that integer k-SUM requires time Ω(n^([k/2]-o(1)). Recent work has studied multidimensional k-SUM in F^d_p, where the best known algorithm takes time O(n^([k/2]). Bhattacharyya et al. [ICS 2011] proved a min(2^(Ω(d)), n^(Ω(k)) lower bound for k-SUM in F^d_p under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F^d_p requires time Ω(n^(k/2-o(1)) if k is even, and Ω(n^([k/2]-2k log k/log p –o(1) if k is odd. For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising O(n^(2−2/(d+13))) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time Ω(n^(2−4/d−o(1))) under the standard 3SUM conjecture, and time Ω(n^(2−2/d−o(1))) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM.

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