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Re: "The Skirting Board Problem"

by blokhead (Monsignor)
on Jan 22, 2008 at 15:06 UTC ( #663596=note: print w/replies, xml ) Need Help??

in reply to "The Skirting Board Problem"

This is obviously an example of the general problem of having a set of numbers and trying to find the minimum number of groups, the sum of whose members does not exceed a certain value.
What you describe is the bin-packing problem. Each length of board represents a bin and the lengths to be cut for each wall represent the set of items to be packed into the fewest number of bins.
I can conceive of a sort of brute force approach of just trying out a whole bunch of combinations, but there must be a more intelligent way.
Not likely, since the problem is NP-complete. On the bright side, there are many simple approximations for this problem which give a guaranteed approximation ratio*. The wikipedia article mentions a few which give quite good ratios (11/9). They use very simple rules. For example: First-fit decreasing would look like this (untested):
my $B = ...; # bucket size my @items = qw[ ... ]; # items my @bins; my @binsizes; ITEM: for my $item (sort { $b <=> $a } @items) { for my $bin (0 .. $#binsizes) { if ($binsizes[$bin] + $item <= $B) { push @{ $bins[$bin] }, $item; $binsizes[$bin] += $item; next ITEM; } } push @bins, [$item]; push @binsizes, $item; } print "[@$_]\n" for @bins;

Other places to look include Algorithm::Bucketizer or Algorithm::BinPack, but I don't know which heuristic algorithm they use under the hood.

BTW, it's likely you have few enough items (with small values) so that the brute-force method will not take too long. But it might take longer to code the brute-force search than to execute it (or install your skirting boards)!

Appendix/Update: A guaranteed approximation ratio of 11/9 means that if the optimal solution to a problem instance uses N bins, then the approximation/heuristic algorithm is guaranteed to return a solution for that instance that uses no more than (11/9)*N bins. (Note that I am ignoring the extra +1 in the guarantee for first-fit decreasing -- it will give you a solution that uses at most (11/9)*N+1 bins)

Also note that this ratio is tight, which means that there are indeed pathological cases where the algorithm gives solutions that really are a (11/9) factor worse than optimal. However, chances are that most inputs will not yield this worst case -- and at least there is a known bound about how bad things can really get.


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