Revisiting the Sparse Matrix Compression Problem

TL;DR

This paper analyzes the computational complexity of sparse matrix compression, showing limitations of greedy heuristics, establishing hardness results, and proposing exact algorithms for specific cases, while also exploring a new variant focused on minimizing array length.

Contribution

It provides a detailed complexity analysis of the sparse matrix compression problem, including approximation bounds, hardness results, and a dynamic programming solution for certain matrix sizes, along with a novel variant minimizing array length.

Findings

Greedy algorithm has an approximation ratio of Θ(√(ℓ+ρ)).

Hardness results for parameterizations like distinct rows and non-zero entries.

DP algorithm solves the problem for double-logarithmic or logarithmic matrix widths.

Abstract

The sparse matrix compression problem asks for a one-dimensional representation of a binary $n \times \ell$ matrix, formed by an integer array of row indices and a shift function for each row, such that accessing a matrix entry is possible in constant time by consulting this representation. It has been shown that the decision problem for finding an integer array of length $\ell+\rho$ or restricting the shift function up to values of $\rho$ is NP-complete (cf. the textbook of Garey and Johnson). As a practical heuristic, a greedy algorithm has been proposed to shift the $i$-th row until it forms a solution with its predecessor rows. Despite that this greedy algorithm is cherished for its good approximation in practice, we show that it actually exhibits an approximation ratio of $\Theta(\sqrt{\ell+\rho})$. We give further hardness results for parameterizations such as the number of distinct rows or the maximum number of non-zero entries per row. Finally, we devise a DP-algorithm that solves the problem for double-logarithmic matrix widths or logarithmic widths for further restrictions. We study all these findings also under a new perspective by introducing a variant of the problem, where we wish to minimize the length of the resulting integer array by trimming the non-zero borders, which has not been studied in the literature before but has practical motivations.