Algorithm to share/settle expenses among a group

I have recently written a blog post describing an approach to solve the settlement of expenses between members of a group where potentially everybody owes everybody else, such that the number of payments needed to settle the debts is the least possible. It uses a linear programming formulation. I also show an example using a tiny R package that implements the solution.

The best way to get back to zero state (minimum number of transactions) was covered in this question here.

I have created an Android app which solves this problem. You can input expenses during the trip, it even recommends you "who should pay next". At the end it calculates "who should send how much to whom". My algorithm calculates minimum required number of transactions and you can setup "transaction tolerance" which can reduce transactions even further (you don't care about $1 transactions) Try it out, it's called Settle Up:

https://market.android.com/details?id=cz.destil.settleup

Description of my algorithm:

I have basic algorithm which solves the problem with n-1 transactions, but it's not optimal. It works like this: From payments, I compute balance for each member. Balance is what he paid minus what he should pay. I sort members according to balance increasingly. Then I always take the poorest and richest and transaction is made. At least one of them ends up with zero balance and is excluded from further calculations. With this, number of transactions cannot be worse than n-1. It also minimizes amount of money in transactions. But it's not optimal, because it doesn't detect subgroups which can settle up internally.

Finding subgroups which can settle up internally is hard. I solve it by generating all combinations of members and checking if sum of balances in subgroup equals zero. I start with 2-pairs, then 3-pairs ... (n-1)pairs. Implementations of combination generators are available. When I find a subgroup, I calculate transactions in the subgroup using basic algorithm described above. For every found subgroup, one transaction is spared.

The solution is optimal, but complexity increases to O(n!). This looks terrible but the trick is there will be just small number of members in reality. I have tested it on Nexus One (1 Ghz procesor) and the results are: until 10 members: <100 ms, 15 members: 1 s, 18 members: 8 s, 20 members: 55 s. So until 18 members the execution time is fine. Workaround for >15 members can be to use just the basic algorithm (it's fast and correct, but not optimal).

Source code:

Source code is available inside a report about algorithm written in Czech. Source code is at the end and it's in English:

https://web.archive.org/web/20190214205754/http://www.settleup.info/files/master-thesis-david-vavra.pdf

You have described it already. Sum all the expenses (1500 in your case), divide by number of people sharing the expense (500). For each individual, deduct the contributions that person made from the individual share (for person A, deduct 400 from 500). The result is the net that person "owes" to the central pool. If the number is negative for any person, the central pool "owes" the person.

Because you have already described the solution, I don't know what you are asking. Maybe you are trying to resolve the problem without the central pool, the "bank"?

I also don't know what you mean by "start with the least spent amount and work forward."