TITLE: Finding a symmetric adjacency matrix closest to a given (non-symmetric) adjacency matrix
QUESTION [6 upvotes]: I am trying to solve a problem on graphs, which I have reduced to the following optimization problem in matrix $X \in \{0,1\}^{n \times n}$
$$\begin{array}{ll} \text{minimize} & \| X - A \|_F^2\\ \text{subject to} & X 1_n = m 1_n\\ & X=X^\top\end{array}$$
where matrix $A \in \{0,1\}^{n \times n}$ is given. Matrix $X$ is the adjacency matrix of a non-directed $m$-regular graph, while matrix $A$ is the adjacency matrix of a directed graph.
I am quite clueless on how to go on solving this problem and would be happy to get a direction.

REPLY [4 votes]: We have the following optimization problem in matrix $\mathrm X \in \{0,1\}^{n \times n}$
$$\begin{array}{ll} \text{minimize} & \| \mathrm  X - \mathrm A \|_{\text{F}}^2\\ \text{subject to} & \mathrm  X 1_n = m 1_n\\ & \mathrm  X = \mathrm  X^\top\\ & \mathrm X \in \{0,1\}^{n \times n}\end{array}$$
where matrix $\mathrm A \in \{0,1\}^{n \times n}$ is given. Note that
$$\| \mathrm  X - \mathrm A \|_{\text{F}}^2 = \| \mathrm  X \|_{\text{F}}^2 - 2 \langle \mathrm  A, \mathrm  X \rangle + \| \mathrm A \|_{\text{F}}^2$$
and that $\| \mathrm  X \|_{\text{F}}^2 = m n$, due to the constraints. Hence, we have the following integer program (IP)
$$\begin{array}{ll} \text{maximize} & \langle \mathrm  A, \mathrm  X \rangle\\ \text{subject to} & \mathrm  X 1_n = m 1_n\\ & \mathrm  X = \mathrm  X^\top\\ & \mathrm X \in \{0,1\}^{n \times n}\end{array}$$
which appears to be a generalization of the assignment problem. Perhaps there is a generalization of the Hungarian algorithm, too.