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

2026-05-10

AI Summary

This study systematically investigates the computational power and limitations of arithmetic circuits over the min-plus semiring—a model commonly used for dynamic programming—in the context of optimization problems. By formally defining the VNP class over this semiring, analyzing the expressive power of constant-width algebraic branching programs (ABPs), and leveraging techniques such as exponential hypercube summation, the work establishes key complexity-theoretic boundaries. The main contributions include a dichotomy theorem based on the number of negations: logarithmically many negations suffice to collapse VNP to VP, whereas super-logarithmically many preserve their separation; a proof that width-3 ABPs are computationally complete while width-2 ABPs cannot solve the minimum-weight 2-edge matching problem; and a demonstration that weak computational models, when augmented with exponential summation, attain full VNP expressiveness.

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 $\mathsf{VNP}$.

Problem

• min-plus semirings
• arithmetic circuit complexity
• VNP
• algebraic branching programs
• dynamic programming

Innovation

• min-plus semiring
• VNP
• algebraic branching programs
• arithmetic circuit complexity
• constant-width ABP