Complex queries for massive data analysis jobs have become increasingly commonplace. Many such queries contain com- mon subexpressions, either within a single query or among multiple queries submitted as a batch. Conventional query optimizers do not exploit these subexpressions and produce sub-optimal plans. The problem of multi-query optimization (MQO) is to generate an optimal combined evaluation plan by computing common subexpressions once and reusing them. Exhaustive algorithms for MQO explore an O(n^n) search space. Thus, this problem has primarily been tackled using various heuristic algorithms, without providing any theoretical guarantees on the quality of their solution. In this paper, instead of the conventional cost minimization problem, we treat the problem as maximizing a linear transformation of the cost function. We propose a greedy algorithm for this transformed formulation of the problem, which under weak, intuitive assumptions, provides an approximation factor guarantee for this formulation. We go on to show that this factor is optimal, unless P = NP. Another noteworthy point about our algorithm is that it can be easily incorporated into existing transformation-based optimizers. We finally propose optimizations which can be used to improve the efficiency of our algorithm.
Submodular maximization has received a significant amount of attention in optimization @cite_3 @cite_28 @cite_4 and has wide applicability in machine learning, computer vision and information retrieval @cite_15 @cite_12 @cite_22 @cite_9 . In this problem, we are given a submodular function @math and a universe @math , with the goal of selecting a subset @math such that @math is maximized. Typically, @math must satisfy additional feasibility constraints such as cardinality, knapsack or matroid constraints.
@cite_4 @cite_22 @cite_28 @cite_9 @cite_3 @cite_15 @cite_12
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We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function @math , and the objective is to find a subset @math maximizing @math . This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by Unconstrained Submodular Maximization include Max-Cut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1 2, thus matching the known hardness result of Feige, Mirrokni, and Vondrak [SIAM J. Comput., 40 (2011), pp. 1133--1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Submodular function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including max cut in digraphs, graphs, and hypergraphs; certain constraint satisfaction problems; maximum entropy sampling; and maximum facility location problems. Our main result is that for any k ≥ 2 and any e > 0, there is a natural local search algorithm that has approximation guarantee of 1 (k + e) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves upon the 1 (k + 1)-approximation of Fisher, Nemhauser, and Wolsey obtained in 1978 [Fisher, M., G. Nemhauser, L. Wolsey. 1978. An analysis of approximations for maximizing submodular set functions---II. Math. Programming Stud.8 73--87]. Also, our analysis can be applied to the problem of maximizing a linear objective function and even a general nonmonotone submodular function subject to k matroid constraints. We show that, in these cases, the approximation guarantees of our algorithms are 1 (k-1 + e) and 1 (k + 1 + 1 (k-1) + e), respectively. Our analyses are based on two new exchange properties for matroids. One is a generalization of the classical Rota exchange property for matroid bases, and another is an exchange property for two matroids based on the structure of matroid intersection. Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a (1 k+2+1 k+e)-approximation for the submodular maximization problem under k matroid constraints, and a (1 5-e)-approximation algorithm for this problem subject to k knapsack constraints (e>0 is any constant). We improve the approximation guarantee of our algorithm to 1 k+1+ 1 k-1 +e for k≥2 partition matroid constraints. This idea also gives a ( 1 k+e)-approximation for maximizing a monotone submodular function subject to k≥2 partition matroids, which improves over the previously best known guarantee of 1 k+1. Submodular function maximization has been studied extensively in recent years under various constraints and models. The problem plays a major role in various disciplines. We study a natural online variant of this problem in which elements arrive one-by-one and the algorithm has to maintain a solution obeying certain constraints at all times. Upon arrival of an element, the algorithm has to decide whether to accept the element into its solution and may preempt previously chosen elements. The goal is to maximize a submodular function over the set of elements in the solution. We study two special cases of this general problem and derive upper and lower bounds on the competitive ratio. Specifically, we design a @math -competitive algorithm for the unconstrained case in which the algorithm may hold any subset of the elements, and constant competitive ratio algorithms for the case where the algorithm may hold at most @math elements in its solution.