Question

I have accounts with positive and negative balance and a pledge relationship between some. A pledge gives accounts with negative balance the right to retrieve money from the pledging account to cover their loss.

I want to find the optimal order of invoking this right of retrieving money.

            1    2    3
A 1000 | -1000 -500 -500
B 1000 | -1000

In the given example account A and B have a positive balance of 1000 and accounts 1,2,3 are covered by priority (1 > 2 > 3). I want to cover as many accounts as possible by distributing the money of A and B on 1,2,3 while honoring the priority.

In this particular example choosing A1 as my first pair would result in only covering 1000 but if I choose B1, A2, A3 I have the optimal solution of covering 2000.

How is this kind of optimization problem called and what are the algorithms to tackle it?

Answer 1

It’s basically a network flow problem. I’ll draw the capacitated graph for your example (unlabeled arcs have infinite capacity). s is the source and t is the sink.

     >A------->1
1000/ |\       ^\
   /  | \     /  \1000
  /   |  \   /    \
 /    |   \ /  500 v
s     |    /->2--->t
 \     \  /        ^
  \     \/        /
   \    /\       /500
1000\  /  \     /
     >B    --->3

The answer isn’t the max flow; it’s the flow that maximizes 1, then 2, then 3. One poly-time algorithm is to modify a max flow algorithm based on augmenting paths (simple paths!—otherwise we might take flow away from a higher priority account) preferentially to augment paths via 1, then 2, then 3.