One feature of your example data here is that the field of interest (the "name" field) is always the first field in the record, i.e., always at the start of a string read from a file. (Update: This approach assumes that the  $rx_sep field separator pattern cannot possibly appear in a "name" field!) This anchor can be very useful. If you build a regex to match all the names "of interest" (see haukex's article Building Regex Alternations Dynamically), it's a one-pass process to read all records in a file and match and extract only those records of interest.

c:\@Work\Perl\monks>perl
use strict;
use warnings;

my @names = qw(1000567/1  1000567/2  122854/1  1000574/2);

my $rx_sep = qr{ , }xms;  # adjust to match real field separator

my ($rx_interesting) =
    map qr{ \A (?: $_) (?= $rx_sep) }xms,
    join q{ | },
    map quotemeta,  # proper!
    reverse sort    # order!
    @names
    ;
print "$rx_interesting \n";  # for debug

my @data = (
  '1083978/2,284224,284292,chrX,255,+,284224,284292,255,0,0,1,68,0',
  '122854/1,284224,284277,chrX,255,+,284224,284277,255,0,0,1,53,0',
  '641613/1,284224,284290,chrX,255,+,284224,284290,255,0,0,1,66,0',
  );

while (my $datum = shift @data) {
    print "interesting: >$datum< \n" if $datum =~ $rx_interesting;
    }

__END__
(?msx-i: \A (?: 122854\/1 | 1000574\/2 | 1000567\/2 | 1000567\/1) (?= 
+(?msx-i: , )) )
interesting: >122854/1,284224,284277,chrX,255,+,284224,284277,255,0,0,
+1,53,0<
[download]

Note that  $rx_sep has to be adjusted to match whatever field separator your data records actually use.

Update 1: I didn't notice that haukex already suggested this approach here. Oh well... At least you have a worked example :)

Update 2: I've noticed a stupid mistake in my code as originally posted. A part of the sequence of operations to build  $rx_interesting was incorrectly given as
    reverse sort
    map quotemeta,
The code has been corrected. The error (quotemeta-ing before sort-ing) should have made no difference in this particular application, but there are corner cases (update: in other potential applications) in which it would (although I'm unable to think of a good example of such a case ATM).