Question

Let’s say I have two sets: (n_1, n_2, …) and (m_1, m_2, …) and a matching function match(n, m) that returns a value from 0 to 1. I want to find the mapping between the two sets such that the following constraints are met:

• Each element must have at most 1 matched element in the opposite set.
• Unmatched elements will be paired with a dummy element at cost 1
• The sum of the match function when applied to all elements is maximal
• I am having trouble expressing this formally, but if you lined up each set parallel to each other with their original ordering and drew a line between matched elements, none of the lines would cross. E.x. [n_1<->m_2, n_2<->m_3] is a valid mapping but [n_1<->m_2, n_2<->m_1] is not.

(I believe the first three are standard weighted bipartite matching constraints, but I specified them in case I misunderstood weighted bipartite matching)

This is relatively straight forward to do with an exhaustive search in exponential time (with respect to the size of the sets), but I’m hoping a polynomial time (ideally O((|n|*|m|)^3) or better) solution exists.

I have searched a fair amount on the “assignment problem”/”weighted bipartite matching” and have seen variations with different constraints, but didn’t find one that matched or that I was able to reduce to one with this added ordering constraint. Do you have any ideas on how I might solve this? Or perhaps a rough proof that it is not solvable in polynomial time (for my purposes, a reduction to NP-complete would also work)?

Answer 1

This problem has been studied under the name “maximum-weight non-crossing matching”. There’s a simple quadratic-time dynamic program. Let M(a, b) be the value of an optimal matching on n₁, …, n_a and m₁, …, m_b. We have the recurrence

M(0, b) = -b
M(a, 0) = -a
M(a, b) = max {M(a – 1, b – 1) + match(a, b), M(a – 1, b) – 1, M(a, b – 1) – 1}.

By tracing back the argmaxes, you can recover an optimal solution from its value.

If match has significantly fewer than a quadratic number of entries greater than -1, there is an algorithm that runs in time linearithmic in the number of useful entries (Khoo and Cong, A Fast Multilayer General Area Router for MCM Designs).