Computing the Best Case Energy Complexity of Satisfying Assignments in Monotone Circuits

Measures of circuit complexity are usually analyzed to ensure the computation of Boolean functions with economy and efficiency. One of these measures is energy complexity, which is related to the number of gates that output true in a circuit for an assignment. The idea behind energy complexity comes from the counting of `firing' neurons in a natural neural network. The initial model is based on threshold circuits, but recent works also have analyzed the energy complexity of traditional Boolean circuits. In this work, we discuss the time complexity needed to compute the best-case energy complexity among satisfying assignments of a monotone Boolean circuit, and we call such a problem as MinEC$^+_M$. In the MinEC$^+_M$ problem, we are given a monotone Boolean circuit $C$, a positive integer $k$ and asked to determine whether there is a satisfying assignment $X$ for $C$ such that $EC(C,X) \leq k$, where $EC(C,X)$ is the number of gates that output true in $C$ according to the assignment $X$. We prove that MinEC$^+_M$ is NP-complete even when the input monotone circuit is planar. Besides, we show that the problem is W[1]-hard but in XP when parameterized by the size of the solution. In contrast, we show that when the size of the solution and the genus of the input circuit are aggregated parameters, the MinEC$^+_M$ problem becomes fixed-parameter tractable.