Monotone Boolean circuits are circuits where each gate is either an AND gate or an OR gate. In other words, negation gates are not allowed on monotone circuits. This class of circuits has sparked the attention of researchers working in several subfields of combinatorics and complexity theory. In this work, we consider the notion of certification-width of a monotone Boolean circuit, a complexity measure that intuitively quantifies the minimum number of edges that need to be traversed by a minimal set of positive weight inputs in order to certify that a given circuit is satisfied. We call the problem of computing this invariant as Succinct Monotone Circuit Certification (SMCC). We prove that SMCC is NP-complete even when the input monotone circuit is planar. Subsequently, we show that k-SMCC, the problem parameterized by the size of the solution, is W[1]-hard, but still in W[P]. In contrast, we show that k-SMCC is fixed-parameter tractable when restricted to monotone circuits of bounded genus.

单调布尔电路是其中每个门都是与门或或门的电路。换句话说，单调电路上不允许使用否定门。这类电路引起了在组合学和复杂性理论的几个子领域工作的研究人员的注意。在这项工作中，我们考虑了单调布尔电路的证明宽度的概念，这是一种复杂性度量，可以直观地量化最小正权重输入集需要遍历的最小边数，以证明给定的电路很满意。我们将计算这个不变量的问题称为简洁单调电路认证 (SMCC)。我们证明SMCC即使输入单调电路是平面的，它也是 NP 完全的。随后，我们证明k -SMCC，由解决方案的大小参数化的问题，是 W[1]-hard，但仍处于 W[P]。相比之下，我们表明当限制于有界属的单调电路时，k -SMCC是固定参数易处理的。