Covering the Relational Join

In this paper, we initiate a theoretical study of what we call the join covering problem. We are given a natural join query instance $Q$ on $n$ attributes and $m$ relations $(R_i)_{i \in [m]}$. Let $J_{Q} = \ \Join_{i=1}^m R_i$ denote the join output of $Q$. In addition to $Q$, we are given a parameter $\Delta: 1\le \Delta\le n$ and our goal is to compute the smallest subset $\mathcal{T}_{Q, \Delta} \subseteq J_{Q}$ such that every tuple in $J_{Q}$ is within Hamming distance $\Delta - 1$ from some tuple in $\mathcal{T}_{Q, \Delta}$. The join covering problem captures both computing the natural join from database theory and constructing a covering code with covering radius $\Delta - 1$ from coding theory, as special cases. We consider the combinatorial version of the join covering problem, where our goal is to determine the worst-case $|\mathcal{T}_{Q, \Delta}|$ in terms of the structure of $Q$ and value of $\Delta$. One obvious approach to upper bound $|\mathcal{T}_{Q, \Delta}|$ is to exploit a distance property (of Hamming distance) from coding theory and combine it with the worst-case bounds on output size of natural joins (AGM bound hereon) due to Atserias, Grohe and Marx [SIAM J. of Computing'13]. Somewhat surprisingly, this approach is not tight even for the case when the input relations have arity at most two. Instead, we show that using the polymatroid degree-based bound of Abo Khamis, Ngo and Suciu [PODS'17] in place of the AGM bound gives us a tight bound (up to constant factors) on the $|\mathcal{T}_{Q, \Delta}|$ for the arity two case. We prove lower bounds for $|\mathcal{T}_{Q, \Delta}|$ using well-known classes of error-correcting codes e.g, Reed-Solomon codes. We can extend our results for the arity two case to general arity with a polynomial gap between our upper and lower bounds.