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 output real solutions.

The algorithm is named after Joseph Fourier and Theodore Motzkin.

Elimination

The elimination of a set of variables, say V, from a system of relations (here linear inequalities) refers to the creation of another system of the same sort, but without the variables in V, such that both systems have the same solutions over the remaining variables.

If all variables are eliminated from a system of linear inequalities, then one obtains a system of constant inequalities. It is then trivial to decide whether the resulting system is true or false. It 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.

Consider a system $S$ of $n$ inequalities with $r$ variables $x_1$ to $x_r$, with $x_r$ the variable to be eliminated. 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 inequalities that are of the form $x_r \geq b_i-\sum_{k=1}^{r-1} a_{ik} x_k$; denote these by $x_r \geq A_j(x_1, \dots, x_{r-1})$, for $j$ ranging from 1 to $n_A$ where $n_A$ is the number of such inequalities;
• those inequalities that are of the form $x_r \leq b_i-\sum_{k=1}^{r-1} a_{ik} x_k$; denote these by $x_r \leq B_j(x_1, \dots, x_{r-1})$, for $j$ ranging from 1 to $n_B$ where $n_B$ is the number of such inequalities;
• those inequalities 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 $4(n/4)^{2^d}$, a double exponential complexity. This is due to the algorithm producing many unnecessary constraints (constraints that are implied by other constraints). The number of necessary constraints grows as a single exponential. Unnecessary constraints may be detected using linear programming.

Imbert's acceleration theorems

Two "acceleration" theorems due to Imbert permit the elimination of redundant inequalities based solely on syntactic properties of the formula derivation tree, thus curtailing the need to solve linear programs or compute matrix ranks.

Define the history $H_i$ of an inequality $i$ as the set of indexes of inequalities from the initial system $S$ used to produce $i$. Thus, $H_i=\{i\}$ for inequalities $i \in S$ of the initial system. When adding a new inequality $k: A_i(x_1, \dots, x_{r-1}) \leq B_j(x_1, \dots, x_{r-1})$ (by eliminating $x_r$), the new history $H_k$ is constructed as $H_k = H_i \cup H_j$.

Suppsoe that the variables $O_k = \{x_{r}, \ldots, x_{r - k + 1}\}$ have been eliminated. Each inequality $i$ partitions the set $O_k$ into:

• $E_i$, the set of effectively eliminated variables. A variable $x_j$ is in the set as soon as at least of inequality in the history $H_i$ of $i$ results from the elimination of x_j.
• $I_i$, the set of implicitely eliminated variables. A variable is implicitely eliminated when it appears in at least one inequality of $H_i$, but appears neither in $i$ nor $E_i$
• all remaining variables.

A non-redundant inequality has the property that its history is minimal.

Theorem (Imbert's first acceleration theorem). If the history $H_i$ of an inequality $i$ is minimal, then $1 + |E_i| \ \leq \ |H_i| \ \leq 1 + \left| E_i \cup (I_i \cap O_k)\right|$.

An inequality that does not satisfy these bounds is necessarily redundant, and can be removed from the system without changing its solution set.

The second acceleration theorem detects minimal history sets:

Theorem (Imbert's second acceleration theorem). If the inequality $i$ is such that $1 + |E_i| = |H_i|$, then $H_i$ is minimal.

This theorem provides a quick detection criterion and is used in practice to avoid more costly checks, such as those based on matrix ranks. See the reference for implementation details.

See also

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

External links

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