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Motivation

I am trying to explore the idea of constraining the approximation of an image represented by an $m$-by-$n$ matrix $A$ by the values on a linearly-spaced interval of $mn$ elements $L$ generated over the unit interval $I=[0,1)$.

Goal

I want to use each element of $L$ once and only once in filling an $m$-by-$n$ matrix $B$. I want to minimize the function

$$f = \sum_{i=0}^m\sum_{j=0}^n{|A_{ij}-B_{ij}|}$$

so each $B_{ij}$ is constrained by the minimization of $f$ and by the bijection induced by its uniqueness in $L$.

Naive Approach

Evaluate the error for all $(mn)!$ choices of $B$.

Question

Computationally-speaking, what is the best algorithm for finding $B$? What is the computational complexity class of that algorithm? How hard is this to do in an approximate way? Is there a greedy algorithm for this?

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  • $\begingroup$ Please define "linearly-spaced interval of $m n$ elements". $\endgroup$ – Rodrigo de Azevedo Mar 21 at 15:48
  • $\begingroup$ Complexity classes are for decision problems, not algorithms. $\endgroup$ – Rodrigo de Azevedo Mar 21 at 15:50

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