Tensor Ranks and the Fine-Grained Complexity of Dynamic Programming

TL;DR

This work develops a unified framework for analyzing dynamic programming problems in high dimensions through tensor-based cost structures. By Introducing kD LWS and 2D LWS^PT, the authors connect DP recurrences to tensor rank and slice rank, yielding both speedups and conditional hardness results. They show polynomial improvements when the cost tensor has constant rank or slice rank 1, but prove near-tight barriers under SETH and APSP for larger ranks or slice ranks, explaining why certain classical problems admit fast algorithms while others do not. The reductions form a hierarchy that links SAT, Min-IP, kMin-IP, and various LWS variants, and they apply to problems like higher-dimensional airplane refueling, nested boxes, matrix-chain multiplication, and optimal BST, thereby unifying known results and deriving new faster algorithms in select cases. Overall, the paper narrows the gap between DP design and fine-grained hardness by tying complexity to the algebraic structure of the cost tensor.

Abstract

Generalizing work of Künnemann, 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.

Figures (7)

• Figure 1: An example polygon triangulation problem. The polygon $P(i,j)$ is partitioned into 3 parts by choosing $k$ and forming a triangle $(i,j,k)$ whose weight is $w_i\cdot w_j\cdot w_k$.
• Figure 2: $\mathsf{2D \; LWS}$. To compute $T[i,j]$, we take the minimum of all possible white circles (plus their respective tensor values $w$) such that their coordinates differ from the target by one coordinate.
• Figure 3: $[\mathsf{Static}] \mathsf{2D \; LWS}$. To calculate $T'[i,j]$ (black circle), we take the minimum over all possible white circles (plus their respective weight values $w$) such that they share all but one coordinate with $T[i,j]$.
• Figure 4: $\mathsf{LWS}$. To compute the value of $T'[j]$ (black circle), we start from $T[0]$ and go through all possible $T[i]$ such that $1 \leq i<j \leq n$ and takes the minimum of all possible values (plus their respective weight values $w$).
• Figure 5: $[\mathsf{Static}] \mathsf{LWS}$. To compute $T'[j]$ (black circle), we take the minimum of all possible white circles from $T[a]$ to $T[a+N-1]$ (plus their respective weight values $w$).

Theorems & Definitions (76)

• Definition 1.1: $\mathsf{kD \; LWS}$
• Definition 1.2
• Definition 1.3: $\mathsf{2D \; LWS}^{\mathsf{PT}}$
• Theorem : \ref{['thm: SAT reduces to kMinIP']}
• Theorem : \ref{['thm: kMinIP to k-1 D LWS']}
• Theorem : \ref{['thm: kLWS to Static kLWS']}
• Theorem : \ref{['thm: static kLWS to static k-1 LWS']}
• Corollary
• Corollary
• Theorem : \ref{['thm: APSP to twoLWS']}