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I was wondering what is the state of the art on approximating the quadratic assignment problem (QAP). In particular, I am interested in the following instance. Suppose the $A = (a_{ij}) \in \{0,1\}^{n \times n}$ is the adjacency matrix of an undirected graph. Let $B = (b_{\ell k}) \in \mathbb{R}^{K \times K}$ and $\pi : [n] \to [K]$ be a map. (We might put some restriction on $\pi$ too, for example, equal number of nodes being mapped to each element of $[K]$.) We would like to solve $$ \arg \max_{\pi} \sum_{ij} a_{ij} b_{\pi(i),\pi(j)} $$ A more specific questions is this: what is known about semi-definite programming relaxations (SDP) of this problem? Are there ones which do not significantly increase the dimension of the problem? Any review papers on recent advancements?

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See the following paper. There may have been other developments since then.

Maximum Quadratic Assignment Problem, Konstantin Makarychev, Rajsekar Manokaran, Maxim Sviridenko, ICALP 2010

http://www.cs.princeton.edu/~kmakaryc/pdf/maxqap.pdf

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