# Average-Case Subset Balancing Problems

Given a set of n input integers, the Equal Subset Sum problem asks us to find two distinct subsets with the same sum. In this paper we present an algorithm that runs in time O^*(3^0.387n) in the average case, significantly improving over the O^*(3^0.488n) running time of the best known worst-case algorithm and the Meet-in-the-Middle benchmark of O^*(3^0.5n). Our algorithm generalizes to a number of related problems, such as the “Generalized Equal Subset Sum” problem, which asks us to assign a coefficient c_i from a set C to each input number x_i such that ∑_i c_i x_i = 0. Our algorithm for the average-case version of this problem runs in time |C|^(0.5-c_0/|C|)n for some positive constant c_0, whenever C={0, ± 1, …, ± d} or {± 1, …, ± d} for some positive integer d (with O^*(|C|^0.45n) when |C|<10). Our results extend to the problem of finding “nearly balanced” solutions in which the target is a not-too-large nonzero offset τ. Our approach relies on new structural results that characterize the probability that ∑_i c_i x_i =τ has a solution c ∈ C^n when x_i's are chosen randomly; these results may be of independent interest. Our algorithm is inspired by the “representation technique” introduced by Howgrave-Graham and Joux. This requires several new ideas to overcome preprocessing hurdles that arise in the representation framework, as well as a novel application of dynamic programming in the solution recovery phase of the algorithm.

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