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I decided to try implement some assignment problem algorithms. I already did some, but I got stuck on the problem described below:

To put it simply, I need to cover all its vertices with the minimum number of non-intersecting simple cycles.

But I don't understand how, does anyone have any ideas? I would be especially glad to see an explanation.

Michael Djokam
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  • Hi! This sounds like the [disjoint cycle cover](https://en.wikipedia.org/wiki/Vertex_cycle_cover) problem. However, I'm not sure about the meaning of "simple" when you say "simple circle". Can you please explain what you mean by "simple"? Do you mean the cycles [shouldn't have chords](https://en.wikipedia.org/wiki/Chordless_cycle)? – Stef May 01 '22 at 18:27
  • As far as I know, simple cycles are closed traversals without revisiting a vertex twice, except for the start and end vertices. – Michael Djokam May 01 '22 at 18:32
  • Please provide enough code so others can better understand or reproduce the problem. – Community May 02 '22 at 01:38

1 Answers1

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This problem is NP-hard via a reduction from the Hamiltonian cycle problem. More specifically, if a graph has a Hamiltonian cycle, then you can cover all the vertices with a single simple cycle, namely the Hamiltonian cycle, and otherwise the graph requires multiple cycles to cover its nodes (if it can even be done at all).

As a result, unless P = NP, there are no polynomial-time algorithms for this problem. You can still solve it using either heuristic searches or brute force, but those approaches won’t necessarily be fast on all inputs.

templatetypedef
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