Re: Is there a name and theory for "arrangement/packing" problems?

In article <sddve25vjks.fsf@xxxxxxxxxxxxxxxxxxxx>,
Chris F Clark <cfc@xxxxxxxxxxxxxxxxxxxx> wrote:
I have 8 reviewers and 24 papers and I would like each paper to be
reviewed by 3 reviewers, but with as many unique combinations of
reviewer pairs as possible. That works out because 24 papers * 3
reviews/paper = 72 reviews = 8 reviewers * 9 reviews/reviewer. Thus,
with each reviewer reviewing 9 papers, the numbers work out perfectly.

There's another number you might want to come out perfectly, which doesn't
quite work in this example. Namely, it would be nice if every pair of
reviewers were to share the same number of papers. Unfortunately, there
are (8 choose 2) = 28 pairs of reviewers, and 24 * (3 choose 2) = 72
opportunities for two reviewers to share the same paper, and 28 does not
exactly divide 72.

If that calculation *had* worked out perfectly, then you would be
looking for a "balanced incomplete block design," for which there is
a huge literature. Since 28 doesn't divide 72, I believe what you're
interested in is a "partially balanced incomplete block design."

I would start by consulting the "CRC Handbook of Combinatorial Designs."
In general, this subject is called "combinatorial design theory" or
simply "design theory" by the pure mathematicians, and "experimental
design" or "design of experiments" by the statisticians and operations
researchers.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
.

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