1) Circulation with demands:

• Each node v has demand d_v
  • d_v > 0 -> sink
  • d_v < 0 -> source
• No infinite source / sink
• Demands must balance
• Goal: Find a feasible circulation

2) Add a lower bound l_e to each edge e:

Idea: create an “initial circulation” that satisfies these lower bounds:

This transforms to:

And finally, inserting start and tail nodes to match the demands on the nodes:

This has a solution with flow = D where D = sum( c_e ) for s -> … iff the original problem has a feasible circulation.

3) Example:

Imagine you have data showing which of n customers have purchases which of k products, and want to send surveys with the following restrictions:

• Customers are only asked about products that they have bought
• Customer i can be asked about c_i to c_i‘ products only
• For each product we want p_i to p_i‘ customer surveys

Is there a way to produce questionaires that satisfy these requirements?

Look for a feasible circulation. There is an integer solution if it exists. Find these integer solutions. For each customer i I create a question product p if there is a flow of l from i to p.

“Claim: The resulting questionaires satisfy all requirements.”

This is obvious from the construction.

“Claim: There is a valid set of surveys iff there exists a feasible circulation.”

Assume we have a valid set of surveys. We can use this to construct a flow on each i -> p edge corresponding to a question. If the surveys are valid, then the circulation rules are satisfied.

To prove the other way, use similar arguments.

This entry was posted on Friday, March 4th, 2005 at 9:53 pm and is tagged with circulation rules, integer solution, integer solutions, customer surveys, infinite source, questionaires, source sink, generalizations, amp, node, pi, lt. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback.