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 $\mathsf{VNP}$.

翻译：算术电路复杂性研究仅使用加法、乘法、减法和除法等算术运算计算多项式所需的复杂性。整数环上的多项式可对计数问题进行建模。类似地，热带半环等半环上的多项式可对优化问题进行建模。半环上的电路则表征所谓的纯算法，即仅使用半环中运算的算法。本文从复杂性理论角度研究半环上电路（代表动态规划）的能力与局限性：i) 我们定义了极小加半环上的$\mathsf{VNP}$，可忠实表征计算最小权完美匹配和最小权哈密顿环等问题，其中存在可高效验证的证书。与环的情况不同，我们可免费补充证书中的值，因为极小加半环上无法实现补运算。我们证明了一个二分定理：若仅补充对数多个值，该类别与极小加半环上的$\mathsf{VP}$等价；若补充超对数多个值，则$\mathsf{VNP} \neq \mathsf{VP}$。ii) 考虑常宽ABP（亦称增量动态规划，限定仅使用常数个寄存器），并证明即使计算最小权2-边匹配这类简单问题也无法用宽度2（即2个寄存器）实现。然而宽度3（即3个寄存器）的程序可计算所有问题。更一般地，我们证明了常深公式可被常宽ABP高效模拟。iii) 我们证明在宽度2 ABP和线性形式积等可证明的弱模型上，指数超立方求和（半环上的极小运算）与$\mathsf{VNP}$等价。