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It has been shown that: it is NP-hard to approximate the maximum 3-dimensional matching problem (Max-3-DM) to within 95/94, this result apply to instances with exactly two occurrences of each element.

Does this mean that, the Max-3-DM with the bound 2 on the number of occurrences of each element in triples, is NP-hard?

I have found a polynomial reduction from the Max-3-DM with bound 2 to my problem, can I say that my problem is NP-hard?

Farah Mind
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2 Answers2

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From what I understand, this sentence means that the approximation problem is NP-hard. It says nothing about Max-3-DM problem itself.

Regardless of that, in order to prove that Your problem is NP-hard, You have to reduce some NP-complete problem to Your problem. So even if Max-3-DM is NP-hard, reduction to Max-3-DM problem is not enough. You would have to reduce Max-3-DM to Your problem (that is, in the opposite direction) and Max-3-DM would have to be NP-complete.

Tony Babarino
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  • In fact I have reduced Max 3-DM with the bound 2 on the number of occurrences of each element in triples to my problem, I mad a mistak in my question. And I would like to know if this is sufficient to say that my problem is NP-hard. I have found that the Max 3-DM with bound bigger than or equal to 3 is NP-hard. But for bound 2 I have found that it is NP-hard to approximate this problem to within 95/94. So what do you think? – Farah Mind Mar 11 '16 at 16:30
  • Well the problem is whether Max-3-DM is NP-complete because NP-hard is not enough. I have found this [paper](ftp://ftp1.nada.kth.se/Theory/Viggo-Kann/max3dm.pdf), which says its MaxSNP-complete. I don't know yet if MaxSNP-complete problems are NP-complete though. – Tony Babarino Mar 11 '16 at 16:39
  • Yes I have read this paper, the complete version can be found [here](http://www.sciencedirect.com/science/article/pii/002001909190246E) . I think it is sufficient to say that Max-3-DM is NP-hard (I read some papers that used the Max-3-DM with bound 3). But my problem is in the bound, in this paper they proved that Max-3-DM is MaxSNP-complete for bound bigger than or equal to 3. But I have used in my reduction bound equal to 2. – Farah Mind Mar 11 '16 at 16:49
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If it is indeed NP-hard to approximate MAX-3DM with exactly two instances of each element within 95/94, then it's NP-hard to solve MAX-3DM with exactly two instances of each element. Specifically, if you could solve the problem exactly, then you could end up with an approximation better than 95/94 of the optimal solution (namely, you'd have something that was exactly accurate).

Generally speaking, if it's NP-hard to approximate a problem to within a factor of 1+ε, it's NP-hard to solve it exactly because an exact solution is essentially a 1-approximation of the true answer.

templatetypedef
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  • it is exactely what I am searching for, I would like just to confirm the following: I have found a polynomial reduction from the Max-3-DM with exactly two instances of each element to my problem, can I say that my problem is NP-hard? – Farah Mind Mar 11 '16 at 17:49
  • I believe so, assuming you're right about the hardness of approximation result. – templatetypedef Mar 11 '16 at 19:26