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

TL;DR

This paper demonstrates that for a broad class of subset sum problems with randomly chosen inputs, the likelihood of requiring exponential time using branch-and-bound algorithms approaches certainty as the problem size grows.

Contribution

It establishes the near-certain exponential complexity of subset sum instances with random data when solved by branch-and-bound, highlighting fundamental hardness.

Findings

Probability of exponential branch-and-bound complexity approaches 1 as n increases.

Instances with randomly chosen inputs are almost surely hard for branch-and-bound.

Hardness persists regardless of variable branching order.

Abstract

Given $n$ positive integers $a_1,a_2,\dots,a_n$, and a positive integer right hand side $\beta$, we consider the feasibility version of the subset sum problem which is the problem of determining whether a subset of $a_1,a_2,\dots,a_n$ adds up to $\beta$. We show that if the right hand side $\beta$ is chosen as $\lfloor r\sum_{j=1}^n a_j \rfloor$ for a constant $0 < r < 1$ and if the $a_j$'s are independentand identically distributed from a discrete uniform distribution taking values ${1,2,\dots,\lfloor 10^{n/2} \rfloor }$, then the probability that the instance of the subset sum problem generated requires the creation of an exponential number of branch-and-bound nodes when one branches on the individual variables in any order goes to $1$ as $n$ goes to infinity.