A Combinatorial Problem Arising From Ecology: the Maximum Empower Problem.

TL;DR: The Maximum Empower Problem can be solved by finding a maximum weighted clique in a cograph, which leads to an exponential-time algorithm in the worst-case, and it is proved that the Maximum Em power Problem is #P-hard in the general case, i.e. it is as hard as computing the permanent of a matrix.

Abstract: The ecologist H T Odum introduced a principle of physics, called Maximum Empower, in order to explain self-organization in a system (eg physical, biological, social, economical, mathematical, ) The concept of empower relies on emergy, which is a second notion introduced by Odum for comparing energy systems on the same basis The roots of these notions trace back to the 50's (with the work of H T Odum and R C Pinkerton) and is becoming now an important sustainability indicator in the ecologist community In 2012, Le Corre and Truffet developed a recursive method, based on max-plus algebra, to compute emergy of a system Recently, using this max-plus algebra approach, it has been shown that the Maximum Empower Principle can be formalized as a new combinatorial optimization problem (called the Maximum Empower Problem) In this paper we show that the Maximum Empower Problem can be solved by finding a maximum weighted clique in a cograph, which leads to an exponential-time algorithm in the worst-case We also provide a polynomial-time algorithm when there is no cycle in the graph modeling the system Finally, we prove that the Maximum Empower Problem is #P-hard in the general case, ie it is as hard as computing the permanent of a matrix