Choose a set of markers to distinguish individuals - r

Question

I have some genotype data from 8 markers and 20 individuals. I would like to select 5 markers from the 8 markers, which can form a unique genotype pattern for each individual. The purpose is to select as few as possible markers to distinguish the 20 individuals.
I know that I need to select 5 columns out of 8, and then compare each row. If we find duplicated rows, then we need to re-select another 5 columns, until we find no duplicated rows.
But I don't know how I can translate it into R. Could somebody help? Thanks!
sample data
Indiv MN1 MN2 MN3 MN4 MN5 MN6 MN7 MN8
1 A C C A C G A T
2 A C T A T A A T
3 A C T G C A A C
4 A C T G C G G C
5 A T C G C A A C
6 A T C G C A G C
7 A T T A T A A T
8 A T T A T A G T
9 A T T A T G G C
10 G C C A C A A C
11 G C C A C G A T
12 G C C G C G G T
13 G C C G T G G T
14 G C T G C G A T
15 G C T G T A G C
16 G T C A T A G T
17 G T C G T A A C
18 G T T A C G G T
19 G T T G T G G T

Answer 1

Impossible. Assume that we can't change the order of markers. You need at least 6 markers to distinguish the individuals. Consider this function (brute-force solution).
distinct_combn <- function(df, m) {
out <- combn(df, m, function(x) {
if (nrow(unique(x)) == nrow(x)) names(x) else character(0L)
}, simplify = FALSE)
out[lengths(out) > 0L]
}
Then we can see that
> distinct_combn(df[, -1L], 5)
list()
> distinct_combn(df[, -1L], 6)
[[1]]
[1] "MN1" "MN2" "MN3" "MN5" "MN6" "MN7"
[[2]]
[1] "MN1" "MN2" "MN3" "MN5" "MN7" "MN8"
[[3]]
[1] "MN1" "MN2" "MN4" "MN5" "MN6" "MN7"
[[4]]
[1] "MN1" "MN2" "MN4" "MN5" "MN7" "MN8"
Data I used
> df
Indiv MN1 MN2 MN3 MN4 MN5 MN6 MN7 MN8
1 1 A C C A C G A T
2 2 A C T A T A A T
3 3 A C T G C A A C
4 4 A C T G C G G C
5 5 A T C G C A A C
6 6 A T C G C A G C
7 7 A T T A T A A T
8 8 A T T A T A G T
9 9 A T T A T G G C
10 10 G C C A C A A C
11 11 G C C A C G A T
12 12 G C C G C G G T
13 13 G C C G T G G T
14 14 G C T G C G A T
15 15 G C T G T A G C
16 16 G T C A T A G T
17 17 G T C G T A A C
18 18 G T T A C G G T
19 19 G T T G T G G T
> dput(df)
structure(list(Indiv = 1:19, MN1 = c("A", "A", "A", "A", "A",
"A", "A", "A", "A", "G", "G", "G", "G", "G", "G", "G", "G", "G",
"G"), MN2 = c("C", "C", "C", "C", "T", "T", "T", "T", "T", "C",
"C", "C", "C", "C", "C", "T", "T", "T", "T"), MN3 = c("C", "T",
"T", "T", "C", "C", "T", "T", "T", "C", "C", "C", "C", "T", "T",
"C", "C", "T", "T"), MN4 = c("A", "A", "G", "G", "G", "G", "A",
"A", "A", "A", "A", "G", "G", "G", "G", "A", "G", "A", "G"),
MN5 = c("C", "T", "C", "C", "C", "C", "T", "T", "T", "C",
"C", "C", "T", "C", "T", "T", "T", "C", "T"), MN6 = c("G",
"A", "A", "G", "A", "A", "A", "A", "G", "A", "G", "G", "G",
"G", "A", "A", "A", "G", "G"), MN7 = c("A", "A", "A", "G",
"A", "G", "A", "G", "G", "A", "A", "G", "G", "A", "G", "G",
"A", "G", "G"), MN8 = c("T", "T", "C", "C", "C", "C", "T",
"T", "C", "C", "T", "T", "T", "T", "C", "T", "C", "T", "T"
)), class = "data.frame", row.names = c(NA, -19L))