VP, VNP and Algebraic Branching Programs over Min-Plus Semirings

TL;DR

This paper explores the computational power and limitations of algebraic circuits over min-plus semirings, introducing new complexity classes, dichotomy results, and showing how circuit width affects computational capabilities.

Contribution

It defines VNP over min-plus semirings, proves a dichotomy theorem based on complement size, and analyzes the computational limits of constant-width algebraic branching programs.

Findings

VNP over min-plus semirings can represent problems like min-weight perfect matchings.

A dichotomy theorem relates complement size to class equality: small complements imply VP, large imply VNP ≠ VP.

Constant-width ABPs have limitations; width 2 cannot compute certain problems, but width 3 can.

Abstract

Arithmetic circuit complexity studies the complexity of computing polynomials using only arithmetic operations such as addition, multiplication, subtraction, and division. Polynomials over rings of integers model counting problems. Similarly, polynomials over semirings such as tropical semirings model optimization problems. Circuits over semirings then model so called pure algorithms, algorithms that only use the operations in the semiring. In this paper, we do a complexity-theoretic study of the power and limitations of circuits (which represent dynamic programs) over semirings: i) We define $\mathsf{VNP}$ over min-plus semirings, which can faithfully represent problems such as computing min-weight perfect matchings and min-weight Hamiltonian cycles where we have efficiently verifiable certificates. Unlike over rings, we complement the values in the certificate for free as complementation is impossible over min-plus semirings. We prove a dichotomy theorem that states that if we only complement logarithmically many values, this class is same as $\mathsf{VP}$ over min-plus semirings. If we complement super-logarithmically many values, then $\mathsf{VNP} \neq \mathsf{VP}$. ii) We consider constant-width ABPs (which are also called incremental dynamic programs that are restricted to use only a constant number of registers) and show that even simple problems like computing the min-weight $2$-edge-matching is impossible with width $2$ (or $2$ registers). However, with width $3$ (or $3$ registers), such programs can compute everything. More generally, we show that constant-depth formulas are efficiently simulated by constant-width ABPs. iii) We show that an exponential hypercube sum (min in the semiring) over even provably weak models such as width-$2$ ABPs and products of linear forms are the same as $\mathsf{VNP}$.