Finding Smallest Witnesses for Conjunctive Queries
International Conference on Database Theory(2023)
摘要
A witness is a sub-database that preserves the query results of the original
database but of much smaller size. It has wide applications in query rewriting
and debugging, query explanation, IoT analytics, multi-layer network routing,
etc. In this paper, we study the smallest witness problem (SWP) for the class
of conjunctive queries (CQs) without self-joins.
We first establish the dichotomy that SWP for a CQ can be computed in
polynomial time if and only if it has {\em head-cluster property}, unless
$\texttt{P} = \texttt{NP}$. We next turn to the approximated version by
relaxing the size of a witness from being minimum. We surprisingly find that
the {\em head-domination} property - that has been identified for the deletion
propagation problem \cite{kimelfeld2012maximizing} - can also precisely capture
the hardness of the approximated smallest witness problem. In polynomial time,
SWP for any CQ with head-domination property can be approximated within a
constant factor, while SWP for any CQ without such a property cannot be
approximated within a logarithmic factor, unless $\texttt{P} = \texttt{NP}$.
We further explore efficient approximation algorithms for CQs without
head-domination property: (1) we show a trivial algorithm which achieves a
polynomially large approximation ratio for general CQs; (2) for any CQ with
only one non-output attribute, such as star CQs, we show a greedy algorithm
with a logarithmic approximation ratio; (3) for line CQs, which contain at
least two non-output attributes, we relate SWP problem to the directed steiner
forest problem, whose algorithms can be applied to line CQs directly.
Meanwhile, we establish a much higher lower bound, exponentially larger than
the logarithmic lower bound obtained above. It remains open to close the gap
between the lower and upper bound of the approximated SWP for CQs without
head-domination property.
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