Reducing Profile-Based Matching to the Maximum Weight Matching Problem

The profile-based matching problem is the problem of finding a matching that optimizes profile from an instance $(G, r, \langle u_1, \dots, u_r \rangle)$, where $G$ is a bipartite graph $(A \cup B, E)$, $r$ is the number of utility functions, and $u_i: E \to \{ 0, 1, \dots, U_i \}$ is utility functions for $1 \le i \le r$. A matching is optimal if the matching maximizes the sum of the 1st utility, subject to this, maximizes the sum of the 2nd utility, and so on. The profile-based matching can express rank-maximal matching \cite{irving2006rank}, fair matching \cite{huang2016fair}, and weight-maximal matching \cite{huang2012weight}. These problems can be reduced to maximum weight matching problems, but the reduction is known to be inefficient due to the huge weights. This paper presents the condition for a weight function to find an optimal matching by reducing profile-based matching to the maximum weight matching problem. It is shown that a weight function which represents utilities as a mixed-radix numeric system with base-$(2U_i+1)$ can be used, so the complexity of the problem is $O(m\sqrt{n}(\log{n} + \sum_{i=1}^{r}\log{U_i}))$ for $n = |V|$, $m = |E|$. In addition, it is demonstrated that the weight lower bound for rank-maximal/fair/weight-maximal matching, better computational complexity for fair/weight-maximal matching, and an algorithm to verify a maximum weight matching can be reduced to rank-maximal matching. Finally, the effectiveness of the profile-based algorithm is evaluated with real data for school choice lottery.