Truly Sub-$3^n$ Min-Sum Subset Convolution and Join Ordering

2026-10-07

AI Summary

This study addresses the longstanding $3^n$ computational bottleneck that constrains Min-Sum subset convolution and database join ordering. To overcome this barrier, we propose an efficient deterministic reduction framework from subset convolution to Min-Plus matrix multiplication. By integrating subcubic algorithms with Las Vegas randomization techniques, this work achieves the first rigorous breakthrough of this complexity bound. Specifically, it attains expected and deterministic time complexities of $O^*(2.9987^n)$ and $O^*(2.9997^n)$, respectively. These results establish a novel theoretical paradigm for combinatorial optimization and query acceleration.

Abstract

We present a deterministic reduction from min-sum subset convolution to min-plus matrix product. We show that if the min-plus product of two $D\times D$ matrices with $β$-bit integer entries can be computed in $D^{3-δ}\operatorname{poly}(β,\log D)$ time for a fixed rational $0<δ<1$, then min-sum subset convolution on an $n$-element universe can be solved in $(2+2^{-δ})^n 2^{O(\sqrt n\log(n+1))}\operatorname{poly}(n,β)$ time. Instantiating this reduction with the recent breakthrough on subcubic min-plus matrix product by Alman and Vassilevska Williams gives a Las Vegas algorithm with expected running time $O^*(2.9987^n)$ and a deterministic algorithm with running time $O^*(2.9997^n)$, strictly breaking the longstanding $3^n$ computational barrier. Notably, these speedups translate directly to database query optimization, yielding the same expected and deterministic running-time bounds for join ordering under the $C_{\mathrm{out}}$ cost function.

Problem

• min-sum subset convolution
• join ordering
• computational barrier
• database query optimization

Innovation

• min-sum subset convolution
• min-plus matrix product
• sub-3^n complexity
• join ordering
• deterministic reduction