I have already asked this question in StackOverflow (open bounty closing on Aug 25th).Let's say I have some finite sets: A, B, ..., KI also have A1, A2, ... An, which are subsets of A; B1, B2, ... Bn, which are subsets of B, etc.Let's say S is the cartesian product A x B x ... x Kand Sn is the cartesian product of An x Bn x ... x KnIs there an algorithm to efficiently determine if the union of all Sn is equivalent to S?Some pointers to literature where I can study this problem are greatly appreciated.
Algorithm to Determine if (Union of Cartesian Products of Subsets) equals (Cartesian Product of Full Sets)
ds.algorithms
The problem is coNP-complete and so unlikely to have a poly-time algorithm.  (I'm sure the observation was made before; I don't know where, but look in Garey-Johnson.)Here is a simple reduction from 3-UNSAT $= \{\psi: \psi$ is an unsatisfiable 3-CNF$\}$.Say $\psi(x_1, \ldots, x_t) = \bigwedge_{j = 1}^m (y_{j, 1}\vee y_{j, 2} \vee y_{j, 3})$, whereeach $y_{j, a}$ is either a variable or a negated variable.In your notation, let $A = B = \ldots = K = \{0, 1\}$, with $t$ such sets (one for each variable).  Think of each element of $S = A^t$ as an assignment to $x_1, \ldots, x_t$.  For each $j \in [m]$, create a Cartesian product set $S_j$ which describes the set of assignments that will falsify the $j^{th}$ clause.  Note that this can easily be done: we restrict precisely the 3 components corresponding to the variables appearing in this clause.The union of the $S_j$'s equals all of $S = \{0, 1\}^t$ iff every assignment to $\psi$ falsifies some clause, i.e., iff $\psi$ is unsatisfiable.