Bipartite maximum matching with added constraints

Question

Suppose you have two lists as follows

List $A$ = $(a_1, a_2, ..., a_m)$

List $B$ = $(b_1, b_2, ..., b_n)$

Each element in list $A$ can be paired with many or no elements in list $B$. I have a function $f(a_i, b_j)$ that returns true or false depending on whether the match is valid or not. However, if you pair element $i$ in list $A$ then you also have to pair the $i$-th element in list $B$. So, for example, the pair can be $(a_i, b_2)$ and $(a_1, b_i)$ or even $(a_i, b_i)$.

I want to find the maximum number of pairings between the two lists with the added constraint of if the $i$-th element of list $A$ is paired, then the $i$-th element of list $B$ must also be paired.

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My intuition says that modifying the maximal matching bipartite algorithm might be useful for this. However, I am stuck in actually figuring out how to do this. Any help would be appreciated.

Answer 1

Let me start by modifying the problem statement to use standard graph theory terminology, according to my interpretation of the original post:

Let $G=(V, E)$ be a bipartite graph, with $V= A \cup B$, with $A=\{a_i\}_{1\leq i \leq m}$, with $B=\{b_j\}_{1 \leq j \leq n}$, and with $E \subseteq A \times B$. For an edge set $S \subseteq E$, let $f(A, S)$ denote the set of integers $i$ such that $(a_i, b_k) \in S$ for some $k$, and let $g(B, S)$ denote the set of integers $j$ such that $(a_k, b_j)\in S$ for some $k$. Find a maximal edge set $S^*\subseteq E$ with the property that $f(A, S^*)\subseteq g(A, S^*)$.

Assuming that this interpretation is correct, this algorithm should work:

Initialize $S = E$. Construct $m$ lists $A_1, A_2, \ldots A_m$, where $A_i = \{j \ |\ (a_i, b_j) \in S\}$, and construct $n$ lists $B_1, B_2, \ldots B_n$, where $B_j = \{i \ |\ (a_i, b_j) \in S\}$. Compute $f(A, S)$ and $g(B, S)$, and take some $c \in f(A, S) - g(B, S)$. Remove all edges of $S$ of the form $(a_c, b_k)$, and update $\{A_i\}, \{B_j\}, f(A,S)$, and $g(B, S)$ accordingly. Repeat until $f(A, S) - g(B, S)$ is empty.

It is easy to see that the resultant $S$ is maximal, as every edge removed was necessary. It is also easy to see that the running time is $O(\lvert V \rvert+\lvert E \rvert)$, as each edge is processed $O(1)$ times.