Processing Succinct Matrices and Vectors

TL;DR

This paper investigates the computational complexity of matrix operations when matrices are represented by multi-terminal decision diagrams (MTDDs) and their extension MTDD+, showing polynomial-time algorithms for some operations but hardness results for others.

Contribution

It introduces MTDD+ as an extension of MTDDs to enable efficient matrix operations and analyzes the complexity of various problems in this framework.

Findings

Accessing entries and matrix multiplication are polynomial-time for MTDD+.

Determinant testing is PSPACE-complete for MTDDs.

Entry computation in matrix products is #P-complete.

Abstract

We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD_+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.