Select N items such that their properties are balanced

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Lets say I have N objects and each of them has associated values A and B. This could be represented as a list of tuples like:

[(3,10), (8,4), (0,0), (20,7),...]

where each tuple is an object and the two values are A and B.

What I want to do is select M of these objects (where M < N) such that the sums of A and B in the selected subset is as balanced as possible. M here is a parameter of the problem, I don't want to find the optimal M. I want to be able to say "give me 100 objects, and make them as balanced as possible".

Any idea if there an efficient algorithm which can solve this problem (not necessarily completely optimally)? I think this might be related to bin-packing, but I'm not really sure.

algorithm
1 Answers

This is a disguised variant of subset-sum. Replace each (A,B) by A-B, and then the absolute value of the sum of all selected A-B values is the "unbalancedness" of the sums. So you really have to select M of those scalars and try to have a sum as close to 0 as possible.

The "variant" bit is because you have to select exactly M items. (I think this is why your mind went to bin-packing rather than subset-sum.) If you have a black-box subset-sum solver you can account for this too: if the maximum single-pair absolute difference is D, replace each (A,B) by (A-B+D) and have the target sum be M*D. (Don't do that if you're doing a dynamic programming approach, of course, since it increases the magnitude of the numbers you're working with.)

Presuming that you're fine with an approximation (and if you're not, you're gonna have a real bad day) I would tend to use Simulated Annealing or Late Acceptance Hill Climbing as a basic approach, starting with a greedy initial solution (iteratively add whichever object results in the minimal difference), and then in each round, considering randomly replacing one object by one not-currently-selected object.

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