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This present paper provides the absolutely necessary corrections to the previous work entitled {\it A polynomial Time Algorithm to Solve The Max-atom Problem} (arXiv:2106.08854v1). The max-atom-problem (MAP) deals with system of scalar inequalities (called atoms or max-atom) of the form: $x \leq a + \max(y,z)$. Where $a$ is a real number and $x,y$ and $z$ belong to the set of the variables of the whole MAP. A max-atom is said to be positive if its scalar $a$ is $\geq 0$ and stricly negative if its scalar $a <0$. A MAP will be said to be positive if all atoms are positive. In the case of non positive MAP we present a saturation principle for system of vectorial inequalities of the form $x \leq A x + b$ in the so-called $(\max,+)$-algebra assuming some properties on the matrix $A$. Then, we apply such principle to explore all non-trivial solutions (ie $\neq -\infty$). We deduce a strongly polynomial method to express all solutions of a non positive MAP. In the case a positive MAP which has always the vector $x^{1}=(0)$ as trivial solution we show that looking for all solutions requires the enumeration of all elementary circuits in a graph associated with the MAP. However, we propose a strongly polynomial method wich provides some non trivial solutions., Comment: in French language

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 $\mu$-calculus., Comment: We thank Tom Van Dijk who pointed out errors in the proposed algorithm. We have proposed another approach in the paper entitled "Looking for all solutions of the Max Atom Problem (MAP)" see: arXiv:2408.14256

In this paper we develop a pure algebraic method which provides an algorithm for testing emptyness of a polyhedron.

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., Comment: 23 pages, 3 figures

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., Comment: Emergy, Graph, Max-plus algebra, Sustainability, Fourth law of thermodynamics

In this paper we remark that Shannon entropy can be expressed as a function of the self-information (i.e. the logarithm) and the inverse of the Lambert $W$ function. It means that we consider that Shannon entropy has the trace form: $-k \sum_{i} W^{-1} \circ \mathsf{ln}(p_{i})$. Based on this remark we define a generalized entropy which has as a limit the Shannon entropy. In order to facilitate the reasoning this generalized entropy is obtained by a one-parameter deformation of the logarithmic function. Introducing a new concept of independence of two systems the Shannon additivity is replaced by a non-commutative and non-associative law which limit is the usual addition. The main properties associated with the generalized entropy are established, particularly those corresponding to statistical ensembles. The Boltzmann-Gibbs statistics is recovered as a limit. The connection with thermodynamics is also studied. We also provide a guideline for systematically defining a deformed algebra which limit is the classical linear algebra. As an illustrative example we study a generalized entropy based on Tsallis self-information. We point out possible connections between deformed algebra and fuzzy logics. Finally, noticing that the new concept of independence is based on t-norm the one-parameter deformation of the logarithm is interpreted as an additive generator of t-norms.

In this paper we show that the so-called array Fr\'echet problem in Probability/Statistics is (max, +)-linear. The upper bound of Fr\'echet is obtained using simple arguments from residuation theory and lattice distributivity. The lower bound is obtained as a loop invariant of a greedy algorithm. The algorithm is based on the max-plus linearity of the Fr\'echet problem and the Monge property of bivariate distribution.