A Combinatorial Problem Arising From Ecology: the Maximum Empower Problem

TL;DR

This paper formalizes the Maximum Empower Problem from ecology as a combinatorial optimization problem, providing algorithms for special cases and proving its computational complexity.

Contribution

It shows that the Maximum Empower Problem can be solved via maximum weighted clique in cographs and proves its #P-hardness in general.

Findings

Maximum weighted clique approach for cographs

Polynomial-time algorithm for acyclic systems

Maximum Empower Problem is #P-hard in general

Abstract

The ecologist H. T. Odum introduced a principle of physics, called Maximum Empower, in order to explain self-organization in a system (e.g. 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, i.e. it is as hard as computing the permanent of a matrix.