Fourier–Motzkin elimination

Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can look for both real and (in some cases) for integer solutions.

The operation is named after Joseph Fourier and Theodore Motzkin.

Elimination

Elimination (or ∃-elimination) of variables V from a system of relations (here, linear inequalities) consists in creating another system of the same kind, but without a subset of variables V, such that both systems have the same solutions over the remaining variables.

If one eliminates all variables from a system of linear inequalities, then one obtains a system of constant inequalities, which can be trivially decided to be true or false, such that this system has solutions (is true) if and only if the original system has solutions. As a consequence, elimination of all variables can be used to detect whether a system of inequalities has solutions or not.

Let us consider a system $S$ of $n$ inequalities with $r$ variables $x_1$ to $x_r$, with $x_r$ the variable to eliminate. The linear inequalities in the system can be grouped into three classes, depending on the sign (positive, negative or null) of the coefficient for $x_r$:

• those that are equivalent to some inequalities of the form $x_r \geq b_i-\sum_{k=1}^{r-1} a_{ik} x_k$; let us note these as $x_r \geq A_i(x_1, \dots, x_{r-1})$, for $i$ ranging from 1 to $n_A$ where $n_A$ is the number of such inequalities;
• those that are equivalent to some inequalities of the form $x_r \leq b_i-\sum_{k=1}^{r-1} a_{ik} x_k$; let us note these as $x_r \leq B_i(x_1, \dots, x_{r-1})$, for $i$ ranging from $n_A+1$ to $n_A+n_B$ where $n_B$ is the number of such inequalities;
• those in which $x_r$ plays no role, grouped into a single conjunction $\phi$.

The original system is thus equivalent to $\max(A_1(x_1, \dots, x_{r-1}), \dots, A_{n_A}(x_1, \dots, x_{r-1})) \leq x_r \leq \min(B_1(x_1, \dots, x_{r-1}), \dots, B_{n_B}(x_1, \dots, x_{r-1})) \wedge \phi$.

Elimination consists in producing a system equivalent to $\exists x_r~S$. Obviously, this formula is equivalent to $\max(A_1(x_1, \dots, x_{r-1}), \dots, A_{n_A}(x_1, \dots, x_{r-1})) \leq \min(B_1(x_1, \dots, x_{r-1}), \dots, B_{n_B}(x_1, \dots, x_{r-1})) \wedge \phi$.

The inequality $\max(A_1(x_1, \dots, x_{r-1}), \dots, A_{n_A}(x_1, \dots, x_{r-1})) \leq \min(B_1(x_1, \dots, x_{r-1}), \dots, B_{n_B}(x_1, \dots, x_{r-1}))$ is equivalent to $n_A n_B$ inequalities $A_i(x_1, \dots, x_{r-1}) \leq B_j(x_1, \dots, x_{r-1})$, for $1 \leq i \leq n_A$ and $1 \leq j \leq n_B$.

We have therefore transformed the original system into another system where $x_r$ is eliminated. Note that the output system has $(n-n_A-n_B)+n_A n_B$ inequalities. In particular, if $n_A = n_B = n/2$, then the number of output inequalities is $n^2/4$.

Complexity

Running an elimination step over $n$ inequalities can result in at most $n^2/4$ inequalities in the output, thus running $d$ successive steps can result in at most $(n/2)^{2^d}$, a double exponential complexity. This is due to the algorithm producing many unnecessary constraints (constraints that are implied by the other ones). The number of necessary constraints grows as a single exponential. Unnecessary constraints may be detected using linear programming.

See also

• Real closed field: the cylindrical algebraic decomposition algorithm performs quantifier elimination over polynomial inequalities, not just linear

References

• Fourier, Joseph (1827). "Histoire de l'Académie, partie mathématique (1824)". 7. Gauthier-Villars. 
• Schrijver, Alexander (1998). Theory of Linear and Integer Programming. John Wiley & sons. pp. 155–156. ISBN 0-471-98232-6. 
• Keßler, Christoph W. "Parallel Fourier–Motzkin Elimination". Universität Trier. CiteSeerX: 10.1.1.54.657. 
• Williams, H. P. (1986). "Fourier's Method of Linear Programming and its Dual". American Mathematical Monthly 93: 681–695. 

External links

• Lectures on Convex Sets, notes by Niels Lauritzen, at Aarhus University, March 2010.