C[omp]ute

From: andrew cooke <andrew@...>

Date: Mon, 29 Aug 2011 17:22:25 -0300

The weather in NY has left me with a free afternoon at work, so I have been
looking at mixed integer programming (this is not completely self-indulgent -
it's related to compressive sensing, which we might use).

My motivating example was this question on Stack Overflow -
http://stackoverflow.com/questions/7076349/is-there-a-good-way-to-do-this-type-of-mining

These notes describe largely separate (because I didn't get very far) aspects:
describing that particular problem and Python libraries.


Python Libraries

It seems that mixed integer and linear programming are not, in general, well
supported by open source projects.  See, for example, the comparison of
running times for commercial and open source solutions here:
http://scip.zib.de/

Support in Python reflects that.  My initial candidate was pulp-or
http://code.google.com/p/pulp-or/ but many links there are dead (with ominous
notes about a "transition").  So then I turned to Coopr
https://software.sandia.gov/trac/coopr which, from the text there, may be what
pulp-or is combining with.  However, the documentation was both minimal and
aimed at experts - I really need something more introductory.

At this point I started looking for an implementation in any language, and
identified SCIP http://scip.zib.de/ (as mentioned earlier) as one of the most
active and efficient.  And - a rare positive moment within this work - it has
a Python 3.2 library that looks very nice
http://code.google.com/p/python-zibopt/ although the installation instructions
suggest things are still rather alpha
http://code.google.com/p/python-zibopt/wiki/Installation


Framing the Problem

At first I assumed that the clustering problem described above would be easy
to solve.  But all my initial approaches contained variable numbers of
variables.  This can be avoided, to some extent, by using control variables,
which are binary flags that indicate whether some other variable is used or
not, but the next problem was how to express various constraints as linear
expressions.

I still don't have a perfect solution, but here are some notes to remind me of
how far I have got:

 - By default, every point is in the vertical group for that column.
 - We want to minimise the number of vertical groups, while also minimising
   "gaps" in horizontal groups.
 - Each point has a vertical (1) / horizontal (0) control variable.  The sum
   of these is one contribution to the final weight.
 - Each point has an additional variable which is the horizontal group
   number.  Originally these start at 0 (left) and increment across the row.
 - The sum of the rightmost group number on each row is also minimised.
   This reduces the number of groups.
 - There is a constraint that each successive group number from left to
   right must be the same as the neighbour to the left, or one more.
 - Finally, there is a complex weight that depends on the number of 
   horizontal "gaps" between two points when they have the same group 
   number.  I think this can be generated programmatically, but it's a bit
   complex (especially since it must be ignored when groups are vertical).
   This weight should also be minimised.

Writing that down, it doesn't seem very impressive, but it was a lot of
work... :o(

Andrew