The problem: Exact Cover by 3-Sets (X3C)

The definition: Given a set X, with |X| = 3q (so, the size of X is a multiple of 3), and a collection C of 3-element subsets of X. Can we find a subset C’ of C where every element of X occurs in exactly one member of C’? (So, C’ is an “exact cover” of X).

The class US (Unique Polynomial-Time) is the class of decision problems solvable by an NP machine such that the answer is 'yes' if and only if exactly one computation path accepts.

UNIQUE-3SAT is the problem of deciding whether the 3CNF Boolean formulas have a unique solution. UNIQUE-3SAT is a known US-complete problem.

Is the version of problem Exact Cover by 3-Sets(UNIQUE-X3C) when the instances have a unique solution a US-complete problem? (which means there is a unique subset C’ as solution)

Thanks in advance!!!

  • $\begingroup$ This sounds very similar to 3-dimensional matching. $\endgroup$ Commented Feb 18, 2017 at 22:40
  • $\begingroup$ I think the right term of this class is UP? $\endgroup$
    – Wei Zhan
    Commented Feb 19, 2017 at 9:14
  • $\begingroup$ @WillardZhan : ​ That's just a different class. ​ ​ ​ ​ $\endgroup$
    – user12859
    Commented Feb 20, 2017 at 0:21
  • $\begingroup$ @RickyDemer Oh, I see. Multiple certificates in this class indicate a 'no' instance. $\endgroup$
    – Wei Zhan
    Commented Feb 20, 2017 at 2:04

1 Answer 1


Yes, it is.

This scanned chapter 3 from Garey-Johnson classic book describe a parsimonious reduction from 3SAT to 3DM. And X3C is a generalization of 3DM. So, the same reduction can be used to parsimoniously reduce 3SAT to 3DM.

Thus, $\mathrm{UNIQUE-3SAT}$ many-one reduces to $\mathrm{UNIQUE-X3C}$. As a consequence, $\mathrm{UNIQUE-X3C}$ is $\mathrm{US}$-complete.


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