Upper bound on the number of steps for solving the subset sum problem by the Branch-and-Bound method

TL;DR

This paper establishes computable upper bounds on the number of steps needed to solve the subset sum problem using a specific Branch-and-Bound approach, aiding in resource estimation.

Contribution

It provides the first explicit upper bounds on the complexity of a particular Branch-and-Bound variant for the subset sum problem, based on input data.

Findings

Upper bounds are easily computed from input data.

Bounds enable preliminary resource estimation.

Applicable to a basic Branch-and-Bound variant.

Abstract

We study the computational complexity of one of the particular cases of the knapsack problem: the subset sum problem. For solving this problem we consider one of the basic variants of the Branch-and-Bound method in which any sub-problem is decomposed along the free variable with the maximal weight. By the complexity of solving a problem by the Branch-and-Bound method we mean the number of steps required for solving the problem by this method. In the paper we obtain upper bounds on the complexity of solving the subset sum problem by the Branch-and-Bound method. These bounds can be easily computed from the input data of the problem. So these bounds can be used for the the preliminary estimation of the computational resources required for solving the subset sum problem by the Branch-and-Bound method.