Efficient Reporting of Top-k Subset Sums

TL;DR

This paper introduces an efficient algorithm for reporting the top-k subset sums with minimized total scores, reducing preprocessing and storage needs, and outperforming existing methods in time and space.

Contribution

It presents a novel approach that constructs a partial metadata structure and incremental subset generation, improving efficiency over prior algorithms for top-k subset sum problems.

Findings

Reduces preprocessing time and space requirements.

Achieves better asymptotic time complexity.

Outperforms existing algorithms in experiments.

Abstract

The "Subset Sum problem" is a very well-known NP-complete problem. In this work, a top-k variation of the "Subset Sum problem" is considered. This problem has wide application in recommendation systems, where instead of k best objects the k best subsets of objects with the lowest (or highest) overall scores are required. Given an input set R of n real numbers and a positive integer k, our target is to generate the k best subsets of R such that the sum of their elements is minimized. Our solution methodology is based on constructing a metadata structure G for a given n. Each node of G stores a bit vector of size n from which a subset of R can be retrieved. Here it is shown that the construction of the whole graph G is not needed. To answer a query, only implicit traversal of the required portion of G on demand is sufficient, which obviously gets rid of the preprocessing step, thereby reducing the overall time and space requirement. A modified algorithm is then proposed to generate each subset incrementally, where it is shown that it is possible to do away with the explicit storage of the bit vector. This not only improves the space requirement but also improves the asymptotic time complexity. Finally, a variation of our algorithm that reports only the top-k subset sums has been compared with an existing algorithm, which shows that our algorithm performs better both in terms of time and space requirement by a constant factor.