fel(1)

💖 NAME 💖

fel — bestie, this one projects an H-representation or a V-representation onto a subset of columns. That's the whole vibe. 🦋

It lives in the lrslib crew, version 7.3, and it's an alias of lrs — we'll get into that.

---

✨ SYNOPSIS ✨

fel [input-file] [output-file]

mpirun -np [procs] mplrs -fel [input-file] [output-file] [option...]

---

🌸 DESCRIPTION 🌸

okay so first thing: fel is an alias of lrs which can also perform these functions via the eliminate and project options. So like, same energy, different door. 🚪✨

🦋 H-representation

The input is projected onto a given set of columns using Fourier-Motzkin elimination. An H-representation is output after each iteration. And listen — mplrs -fel is a fully parallel version that performs one iteration only, so go see mplrs(1) for that one. 👑

🎀 V-representation

The specified columns are extracted and redundant rows are removed. The output can be can be piped directly into lrs. 🫶

Not implemented in mplrs. Just so you know, bestie. 😭

---

💅 OPTIONS 💅

These options are placed after the end statement. Yes, after. Not before. iykyk. 🎀

• eliminate k i_1 i_2 ... i_k — (new in v7.2) 🆕
  • (H-representation) Eliminates k variables in an H-representation corresponding to cols i_1 .. i_k by projection onto the remaining variables using the Fourier-Motzkin method. Variables are eliminated in the order given and redundancy is removed after each iteration. slay.
  • (V-representation) Delete the k given columns from the input matrix and remove redundancies (cf. extract where redundancies are not removed). 🫣
  • Column indices are between 1 and n-1 and column zero cannot be eliminated. The output is a valid lrs input file. 💖
• project k i_1 i_2 ... i_k — (new in v7.2) 🆕
  • (H-representation) Project the polyhedron onto the k variables corresponding to cols i_1 .. i_k using the Fourier-Motzkin method. Column indices are between 1 and n-1 and column zero is automatically retained. Variables not contained in the list are eliminated using a heuristic which chooses the column which minimizes the product of the number of positive and negative entries. Redundancy is removed after each iteration using linear programming. 🧠✨
  • (V-representation) Extract the k given columns from the input matrix and remove redundancies. Column indices are between 1 and n-1 and column zero is automatically extracted (cf. extract where redundancies are not removed).
  • The output as a valid lrs input file. 💫

See also extract in lrs(1).

• verbose — For an H-representation give, for each iteration, statistics on how many new inequaties were created and how many of these were redundant. See Example below. 📊

---

🌈 EXAMPLE 🌈

(1) Project onto columns 1 and 3 in an H-representation

 % cat cube.ine
cube.ine
H-representation
begin
6 4 rational
1  1  0  0
1  0  1  0
1  0  0  1
1 -1  0  0
1  0  0 -1
1  0 -1  0
end
project 2 1 3
verbose

 % fel cube.ine
fel:lrslib_v.7.3_2024.1.10(64bit,lrslong.h,hybrid_arithmetic)
*Input taken from  cube.ine
cube

*switching to fel mode
*project 2  1 3
*after removing column 2
*checking for redundant rows only
H-representation
begin
4 3 rational
 0  1  0
 1 -1  0
 1  0 -1
-1  0  1
end
*number of       number after    number of       remaining      column
*inequalties     removing col    redundancies     rows          removed
*         7              6             2             4             2
--------------------------------------------------------------------

*original vars remaining: 1 3

(2) Eliminate columns 1,2,3 from a V-representation

 % cat cp4.ext
V-representation
begin
8 7 rational
 1  0  0  0  0  0  0
 1  0  1  1  1  1  0
 1  1  1  0  0  1  1
 1  1  0  0  1  1  0
 1  0  0  1  0  1  1
 1  1  1  1  0  0  0
 1  0  1  0  1  0  1
 1  1  0  1  1  0  1
end
eliminate 3 1 2 3

 % lrs cp4.ext
lrs:lrslib_v.7.3_2024.1.12(64bit,lrslong.h,hybrid_arithmetic)
*Input taken from  cp4.ext
cp4

*switching to fel mode
*eliminate 3  1 2 3
*checking for redundant rows only
V-representation
begin
4 4 rational
 1  1  1  0
 1  0  1  1
 1  0  0  0
 1  1  0  1
end
*columns retained: 0 4 5 6

---

🎀 NOTES 🎀

1. FAQ page https://inf.ethz.ch/personal/fukudak/polyfaq/polyfaq.html 🌐
2. User's guide for lrslib http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html 📖

---

👑 AUTHOR 👑

David Avis <avis at cs dot mcgill dot ca > 💌

---

🫶 SEE ALSO 🫶

lrs(1), mplrs(1), lrslib(5),

---

okay bestie that's the whole page, mwah 💋 this was lrslib 7.3, dropped January 2024, version tag 2024.1.10, and honestly? it ate. go project something gorgeous. ✨🦋💅