The problem smells badly to being an NP-complete problem. And those are believed that they cannot be done efficiently. Hence, we might as well backtrack. And which mechanism in Perl is good at backtracking? Right, regular expressions.

Therefore, I present a solution that will use a regular expression to do the hard work. It will report the minimum height that is needed. Calculating the actual partition is left as an exercise for the reader.

Abigail

#!/usr/bin/perl

use strict;
use warnings 'all';

sub partition ($$);

my $columns = 4;
my @sizes   = qw /10 15 25 30 10 13/;
    
    
sub partition ($$) {
    my ($b, $h) = @_;
      
    return [(0) x $h] unless $b;
    return [$b]       if $h < 2;   

    map {my $__ = $_; map {[$__ => @$_]} partition $b - $__, $h - 1} 0
+ .. $b;
}
    
my @r = partition @sizes, $columns; 
        
my @regex;
foreach my $r (@r) {
    my $c = 0;
    push @regex => join ":" => map {
        $c += my $__ = $_;
        join ("" => map {"-" x $_} @sizes [$c - $__ .. $c - 1]) . "-*"
+} @$r;
}

my $regex = join "|" => map {"(?:$_)"} @regex;
    
my $try = 1;

{   exit !print "Minimum required height: $try\n" if
          join (":" => map {"-" x $try} 1 .. $columns) =~ /$regex/;
    $try ++;
    redo;
}
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