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Let $m$ and $n$ be two integers, where $m \leq n$. Suppose you are given $m^2$ matrices $W^{i,j} \in \mathbb{R}^{n \times n}$ for $i, j \in \{1, \dots, m\}$.

The goal is to find a sequence $a$ of $m$ integers $a_i \in \{1, \dots, n\}$ minimizing

$$ \min_a \sum_{i=1}^m \sum_{j=1}^m W^{i,j}_{a_i, a_j} $$

Does this problem have an established name?

The total enumeration takes $n^m$ steps. I'm interested in $m \approx 10$ and $n \approx 100$, if not larger. So I wonder if there are better algorithms that should work here aside from simulated annealing and genetic algorithms?

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  • $\begingroup$ This seems to be a long shot, but you could try to reduce SAT to this problem in such a way that SETH-conjecture implies the time lower-bound you specified (assuming negative results). For that given a formula of $N$ variables, you should build an instance such that $m \log n = N$. $\endgroup$ Jan 5, 2022 at 22:18
  • $\begingroup$ Are you looking for an algorithm that gives the exact optimum, or heuristics that give a pretty-good solution but may or may not give the optimal solution? $\endgroup$
    – D.W.
    Jan 5, 2022 at 22:59
  • $\begingroup$ @D.W. Good heuristics, I guess, assuming that exact algorithms are inefficient ($n^m$ steps) $\endgroup$
    – MWB
    Jan 5, 2022 at 23:09
  • $\begingroup$ Is there any structure to W at all? $\endgroup$
    – TickaJules
    Jan 5, 2022 at 23:13
  • $\begingroup$ @TickaJules For the purposes of this question, $W$s are arbitrary. $\endgroup$
    – MWB
    Jan 5, 2022 at 23:15

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