Selective algorithm processing of subset sum distributions

TL;DR

The paper tackles the exact subset sum problem by introducing a bin-based selective algorithm that switches among additive DP, subtractive DP, and repeated-value DP per bin and input value. By mapping the $T$-range into $kn$ bins and tracking computed versus uncomputed sums, it aims to minimize duplicates and balance local efficiencies, achieving an empirical time complexity of $O\left(\max(T, n^2)\right)$ and space $O\left(\max(T, n)\right)$ for $k=2$. Key innovations include bin-wise efficiency estimation, adaptive partitioning, and the new DP variants (SDP, RVDP) along with IPSCD for handling common-divisor-heavy inputs. The results show robust performance across broad problem sizes, with explicit handling for worst-case scenarios and practical guidance on bin design, though a formal proof remains an open theoretical question. Overall, the approach provides a practical, scalable framework for improving exact subset sum computations in diverse distributions.

Abstract

The efficiency of exact subset sum problem algorithms which compute individual subset sums is defined as $e=min(T/z, 1)$, where $z$ is the number of subset sums computed. $e$ is related to these algorithms' computational complexity. This system maps the sums into $kn$ bins to select its most efficient algorithm for each bin for each input value. These algorithms include additive, subtractive and repeated value dynamic programming. Cases which would otherwise be processed inefficiently (eg: all even values) are handled by modular arithmetic and by dynamically partioning the input values. The system's experimentally validated efficiency corresponds to O(max($T$, $n^2$)) with space complexity O(max($T$, $n$)), for $k=2$.