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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.

Rodrigo de Azevedo
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Jonathan
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  • Are all those conditions saying that $X$ is adjacency matrix or am I missing some subtlety there? – Radost Waszkiewicz Jun 04 '19 at 09:59
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    X is an adjacency matrix for a non directed graph, where each vertex is of rank m. While the input A is an adjacency matrix of a directed graph. – Jonathan Jun 04 '19 at 10:08
  • Thanks, that's much clearer! – Radost Waszkiewicz Jun 04 '19 at 10:08
  • I'm guessing $||.||_F$ is Frobenius norm. And the min has to be taken along the undirected graph otherwise the obvious solution would be $X=A$. – Radost Waszkiewicz Jun 04 '19 at 10:11
  • The norm is Frobenius. As for the second observation, $X=A$ would also not hold the constraint of the row sums, as we have no information about them in A. – Jonathan Jun 04 '19 at 10:15
  • Having thought about this, this is a case of integer quadratic programming (albeit that is an enormous class of problems) for sure but this doesn't have any efficient solutions (some algorithms for finding correct ones exist). I feel like there is a lot bigger chance at finding a non-terrible algorithm looking at the original graph problem. On the other hand if you're quite sure that no efficient algorithm exist maybe there is a chance to show that there is a large class of QP problems that are equivalent. – Radost Waszkiewicz Jun 04 '19 at 18:31
  • If $A=|a_{ij}|$ then is it necessary that at most one of $a_{ij}$ and $a_{ij}$ equals $1$ for each $i,j$? – Alex Ravsky Jun 07 '19 at 02:09

1 Answers1

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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.

Rodrigo de Azevedo
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