Average-Case Subset Balancing Problems

by   Xi Chen, et al.

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.


page 1

page 2

page 3

page 4


Equal-Subset-Sum Faster Than the Meet-in-the-Middle

In the Equal-Subset-Sum problem, we are given a set S of n integers and ...

Subset Sum in Time 2^n/2 / poly(n)

A major goal in the area of exact exponential algorithms is to give an a...

Classical and quantum dynamic programming for Subset-Sum and variants

Subset-Sum is an NP-complete problem where one must decide if a multiset...

A complete anytime algorithm for balanced number partitioning

Given a set of numbers, the balanced partioning problem is to divide the...

The Essential Algorithms for the Matrix Chain

For a given product of n matrices, the matrix chain multiplication probl...

On the Hardness of Almost All Subset Sum Problems by Ordinary Branch-and-Bound

Given n positive integers a_1,a_2,...,a_n, and a positive integer right ...

Analysis of Nederlof's algorithm for subset sum

In 2017, J. Nederlof proposed an algorithm [Information Processing Lette...

Please sign up or login with your details

Forgot password? Click here to reset