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Cardinality estimation and conjunctive query evaluation are two of the most fundamental problems in database query processing. Recent work proposed, studied, and implemented a robust and practical information-theoretic cardinality estimation framework. In this framework, the estimator is the cardinality upper bound of a conjunctive query subject to ''degree-constraints'', which model a rich set of input data statistics. For general degree constraints, computing this bound is computationally hard. Researchers have naturally sought efficiently computable relaxed upper bounds that are as tight as possible. The polymatroid bound is the tightest among those relaxed upper bounds. While it is an open question whether the polymatroid bound can be computed in polynomial-time in general, it is known to be computable in polynomial-time for some classes of degree constraints. Our focus is on a common class of degree constraints called simple degree constraints. Researchers had not previously determined how to compute the polymatroid bound in polynomial time for this class of constraints. Our first main result is a polynomial time algorithm to compute the polymatroid bound given simple degree constraints. Our second main result is a polynomial-time algorithm to compute a ''proof sequence'' establishing this bound. This proof sequence can then be incorporated in the PANDA-framework to give a faster algorithm to evaluate a conjunctive query. In addition, we show computational limitations to extending our results to broader classes of degree constraints. Finally, our technique leads naturally to a new relaxed upper bound called theflow bound,which is computationally tractable. 

Datalog^ois an extension of Datalog that allows for aggregation and recursion over an arbitrary commutative semiring. Like Datalog, Datalogo programs can be evaluated via the natural iterative algorithm until a fixed point is reached. However unlike Datalog, the natural iterative evaluation of some Datalogo programs over some semirings may not converge. It is known that the commutative semirings for which the iterative evaluation of Datalogo programs is guaranteed to converge are exactly those semirings that are stable. Previously, the best known upper bound on the number of iterations until convergence over p-stable semirings is ∑i=1 ^n (p+2)ⁱ= Θ(pⁿ) steps, where n is (essentially) the output size. We establish that, in fact, the natural iterative evaluation of a Datalogo program over a p-stable semiring converges within a polynomial number of iterations. In particular our upper bound is O(σ p n²( n²lg Λ + lg σ)) where σ is the number of elements in the semiring present in either the input databases or the Datalogo program, and λ is the maximum number of terms in any product in the Datalogo program. 

Abstract This paper considers the basic problem of scheduling jobs online with preemption to maximize the number of jobs completed by their deadline onmidentical machines. The main result is anO(1) competitive deterministic algorithm for any number of machines$$m >1$$ $m>1$.