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$3$-$\mathrm{Partition}$ problem is $\mathsf{NP}$-Complete in a strong sense meaning there is no pseudo-polynomial time algorithm for it unless $\mathsf{P=NP}$. I am looking for the fastest known exact algorithm that solves $3$-$\mathrm{Partition}$.

Is there a fast (e.g subexponential) algorithm for $3$-$\mathrm{Partition}$? Is it possible to solve it faster than using SAT solvers?

Thinh D. Nguyen
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  • If this does not get a good answer after a few days, it's a candidate for migration to [cstheory.SE]. – Raphael Mar 26 '13 at 13:01
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    @Raphael, Yes I agree, but I didn't search enough but by my knowledge I couldn't find anything, so I said it's better to first ask it here. –  Mar 26 '13 at 13:23
  • Clearly you need to go with an approximation algorithm and it will have to be at best a PTAS (if that is achievable).

    I have not come across any specific approximations for 3-partition, but if you would like to construct your own, there are a couple of standard tricks you could use to construct it (works for any kind of approximation) -- (1) Take the DP solution and approximate the state-space down to some polynomial size, or (2) Perform input rounding so that the inputs produce a polynomial size state-space, or (3) a combination of (1) and (2). It would be cool to see what bounds you get.

     – RDN Mar 26 '13 at 19:32
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    @RDN, First I'm not looking for apprximation, approximation is another problem, second, I cannot see how DP is helpful, I strongly mentioned this is strong NP-Compelete, I don't think you could find any good DP for this (except converting to bin packing and allowing 3 item per bin, which is not really DP). –  Mar 27 '13 at 09:16
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    @RDN Note that "subexponential" is not (strictly) the same as "polynomial", see here. I guess Saeed would also be interested in an algorithm that runs in time, say, $O(1.1^n)$. – Raphael Mar 27 '13 at 12:09
  • @Raphael, actually subexponential is very nice, but yes anything better than SAT solvers is nice. Actually I'm working in context of parametrized complexity and subexponential algorithms are very nice for me. –  Mar 27 '13 at 12:53
  • Erm, you can get sub-exponential with DP by just collapsing the high dimensional state space into a lower dimensional space. I think there are papers by Dr. Gerhard Woeginger on how to do this for both problems that are weakly NP-complete and also ones that are strongly Np-complete.

    I don't see the issue with what I said. And if you don't want an approximation and just want heuristics, well there's probably many of those.

     – RDN Mar 27 '13 at 14:06
  • @RDN, Please see my question I neither asked about approximation nor about heuristics nor anything else except exact algorithm (I think by saying subexponential algorithm is clear that I'm looking for exact algorithm), but, I didn't see the paper you said also I couldn't understand the dynamic programing you talking about, if you know something interesting to answering my question, is nice of you to share it with exact reference and descriptions. P.S: I edited my question to say I'm looking for just exact algorithm. –  Mar 27 '13 at 15:52
  • Even if there was an exact algorithm, it might be that it is not too practical. It might be really hard to beat a SAT solver. Also, if you actually need to solve the problem, have you tried formulating it as SAT and using a solver? – Juho Mar 27 '13 at 16:15
  • @Juho, I don't want to solve practical algorithm, This comes from pure theoretical problem, actually by mentioning SAT solvers, I want to say do not converting to any SAT problem, or proving there is no way faster than using SAT algorithms, really I'm not looking for fast algorithm to get practical results, I'm just looking for theoretically fast algorithm, sure in practice everyone prefers heuristic and approximations. –  Mar 27 '13 at 16:38

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If you find a fast (sub-exponential) exact algorithm for 3-Partition , while 3-Partition is NP-Complete, then dear sir you have just proven P=NP, which no body has done so far, and you have a 1M $ price waiting for you as well as wealth and fame, cheers.

Anyway if you do have a faster than a SAT solver, wouldn't you actually be performing SAT it's self faster? I mean if you can reduce SAT to 3-PARTITION and you can solve 3-PARTITION in sub-expo time, doesn't that mean that SAT can be solved in sub-expo time, hence all NP-COMPLETE problems also can be solved in such time? So you might as well be looking for a sub-expo time solution of any other NP-COMPLETE problem just as well, it would still be a significant breakthrough.

Anwar Saiah
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