Substitution for minimizing/maximizing a tropical linear (fractional) programming

Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state of such game leads to win the game is known to be equivalent to solve a tropical linear optimization problem. This paper mainly focus on the tropical linear minimization problem using a special substitution method on the tropical cone obtained by homogenization of the initial tropical polyhedron. But due to a particular case which can occur in the minimization process based on substitution we have to switch on a maximization problem. Nevertheless, forward-backward substitution is known to be strongly polynomial. The special substitution developed in this paper inherits the strong polynomiality of the classical substitution for linear systems. This special substitution must not be confused with the exponential execution time of the tropical Fourier-Motzkin elimination. Tropical fractional minimization problem with linear objective functions is also solved by tropicalizing the Charnes-Cooper’s transformation of a fractional linear program into a linear program developed in the usual linear algebra. Let us also remark that no particular assumption is made on the polyhedron of interest. Finally, the substitution method is illustrated on some examples borrowed from the litterature.

💡 Research Summary

This paper presents a novel substitution method for solving tropical (max, +) linear programming problems, with a primary focus on minimization. Tropical polyhedra are central to applications in software static analysis and game theory, particularly in determining winning strategies for mean payoff and energy games, which are equivalent to solving tropical linear optimization problems.

The core contribution is an algorithmic framework that solves the tropical linear minimization problem min c^T ⊗ x subject to constraints A ⊗ x ⊕ b ≥ C ⊗ x ⊕ d. The method begins by homogenizing the polyhedron, introducing a variable h > O to form a cone C(A+, A-). A key trick is the application of Fourier’s method to reformulate the minimization as a problem of finding the minimum z subject to the constraint z ≥ c^T ⊗ x ⊕ c_h ⊗ h alongside the original constraints.

The substitution algorithm operates on this cone representation. It iteratively eliminates variables x_j from the problem. At each step, a relevant variable x_j* is identified based on a hierarchy among variables (dominating variables) and a hierarchy among linear functions (h-bounded vs. h-unbounded). For the selected variable, a “valid inequality” of the form a+_i,j ⊗ x_j ≥ a-_i,. ⊗ w is used to express x_j as a linear function of the other variables and h, i.e., x_j = f(x_other, h). This expression is then substituted into all other constraints and the objective bound, effectively reducing the problem dimension by one. Theorems 4.2 and 4.3 ensure that this process maintains optimality. The process repeats until all x variables are eliminated, leaving only constraints on z and h. The final solution for z is then derived from the simplified constraints on h.

A notable feature is the method’s flexibility: during minimization, if the objective depends only on h and remaining variables have upper bounds but no lower bounds, the algorithm can switch to solving a dual maximization problem to find the optimum. The authors prove that this specialized forward-backward substitution inherits the strong polynomial-time complexity of classical linear substitution, distinguishing it from the exponential-time tropical Fourier-Motzkin elimination.

Furthermore, the paper extends the method to tropical linear fractional programming. By “tropicalizing” the classical Charnes-Cooper transformation, a fractional objective is converted into an equivalent linear form, allowing the same substitution method to be applied without additional assumptions on the polyhedron.

The complexity of the overall method is estimated to be roughly O(n^2 m). The paper concludes with illustrative numerical examples and a discussion comparing the approach to existing works, positioning it as an efficient and general algorithmic framework for tropical linear optimization.