Tensor Ranks and the Fine-Grained Complexity of Dynamic Programming

TL;DR

This paper investigates the complexity of high-dimensional dynamic programming problems by analyzing tensor ranks, revealing when polynomial speedups are possible and establishing tight complexity bounds based on tensor properties.

Contribution

It introduces new algorithms and complexity results for DP problems based on tensor rank and slice rank, extending prior work and providing a detailed fine-grained complexity analysis.

Findings

Polynomial speedup possible with constant tensor rank or slice rank 1

Speedup impossible if tensor rank is super-constant (assuming SETH)

Complexity characterized for many classical problems based on tensor properties

Abstract

Generalizing work of K\"unnemann, Paturi, and Schneider [ICALP 2017], we study a wide class of high-dimensional dynamic programming (DP) problems in which one must find the shortest path between two points in a high-dimensional grid given a tensor of transition costs between nodes in the grid. This captures many classical problems which are solved using DP such as the knapsack problem, the airplane refueling problem, and the minimal-weight polygon triangulation problem. We observe that for many of these problems, the tensor naturally has low tensor rank or low slice rank. We then give new algorithms and a web of fine-grained reductions to tightly determine the complexity of these problems. For instance, we show that a polynomial speedup over the DP algorithm is possible when the tensor rank is a constant or the slice rank is 1, but that such a speedup is impossible if the tensor rank is slightly super-constant (assuming SETH) or the slice rank is at least 3 (assuming the APSP conjecture). We find that this characterizes the known complexities for many of these problems, and in some cases leads to new faster algorithms.