arXiv subjects

Chams Lahlou

Publications and source records attributed to Chams Lahlou.

3 recordsLinked to original sources

arXiv2024

A polynomial Time Algorithm to Solve The Max-atom Problem

In this paper we consider $m$ ($m \geq 1$)conjunctions of Max-atoms that is atoms of the form $\max(z,y) + r \geq x$, where the offset $r$ is a real constant and $x,y,z$ are variables. We show that the Max-atom problem (MAP) belongs to $\textsf{P}$. Indeed, we provide an algorithm which solves the MAP in $O(n^{6} m^{2} + n^{4} m^{3} + n^{2} m^{4})$ operations, where $n$ is the number of variables which compose the max-atoms. As a by-product other problems also known to be in $\textsf{NP} \cap \textsf{co-NP}$ are in $\textsf{P}$. P1: the problem to know if a tropical cone is trivial or not. P2: problem of tropical rank of a tropical matrix. P3: parity game problem. P4: scheduling problem with AND/OR precedence constraints. P5: problem on hypergraph (shortest path). P6: problem in model checking and $μ$-calculus.

math.CO↗

arXiv2019

Self-organization and the Maximum Empower Principle in the Framework of max-plus Algebra

Self-organization is a process where order of a whole system arises out of local interactions between small components of a system. Emergy, spelled with an 'm', defined as the amount of (solar) energy used to make a product or service, is becoming an important ecological indicator. The Maximum Empower Principle (MEP) was proposed as the fourth law of thermodynamics by the ecologist Odum in the 90's to explain observed self-organization of energy driven systems. But this principle suffers a lack of mathematical formulation due to an insufficiency of details about the underlying computation of empower (i.e. emergy per time). For empower computation in steady-state an axiomatic basis has been developed recently by Le Corre and the second author of this paper. In this axiomatic basis emergy is defined as a recursive max-plus linear function. Using this axiomatic basis and a correspondance between ecological theory and dynamic systems theory, we prove the MEP. In particular, we show that the empower computation in steady-state is equivalent to a combinatorial optimization problem.

cs.FL↗

arXiv2018

A Combinatorial Problem Arising From Ecology: the Maximum Empower Problem

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.

cs.DM↗