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.
This problem is @math -hard even for the simplest problems which involve only and monotone functions. @cite_28 show that a simple greedy algorithm gives a @math approximation for monotone submodular maximization under cardinality constraints. They further show that it is @math -hard to obtain a better approximation guarantee. Sviridenko @cite_6 presents a modified greedy algorithm for monotone submodular function maximization under knapsack constraints. Their algorithm is the main motivation for our marginal greedy algorithm.
@cite_28 @cite_6
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We consider the problem of maximizing a (non-monotone) submodular function subject to a cardinality constraint. In addition to capturing well-known combinatorial optimization problems, e.g., Max-k-Coverage and Max-Bisection, this problem has applications in other more practical settings such as natural language processing, information retrieval, and machine learning. In this work we present improved approximations for two variants of the cardinality constraint for non-monotone functions. When at most k elements can be chosen, we improve the current best 1 e -- o(1) approximation to a factor that is in the range [1 e + 0.004, 1 2], achieving a tight approximation of 1 2 -- o(1) for k = n 2 and breaking the 1 e barrier for all values of k. When exactly k elements must be chosen, our algorithms improve the current best 1 4 -- o(1) approximation to a factor that is in the range [0.356, 1 2], again achieving a tight approximation of 1 2 -- o(1) for k = n 2. Additionally, some of the algorithms we provide are very fast with time complexities of O(nk), as opposed to previous known algorithms which are continuous in nature, and thus, too slow for applications in the practical settings mentioned above. Our algorithms are based on two new techniques. First, we present a simple randomized greedy approach where in each step a random element is chosen from a set of "reasonably good" elements. This approach might be considered a natural substitute for the greedy algorithm of Nemhauser, Wolsey and Fisher [45], as it retains the same tight guarantee of 1--1 e for monotone objectives and the same time complexity of O(nk), while giving an approximation of 1 e for general non-monotone objectives (while the greedy algorithm of Nemhauser et. al. fails to provide any constant guarantee). Second, we extend the double greedy technique, which achieves a tight 1 2 approximation for unconstrained submodular maximization, to the continuous setting. This allows us to manipulate the natural rates by which elements change, thus bounding the total number of elements chosen. Is it possible to maximize a monotone submodular function faster than the widely used lazy greedy algorithm (also known as accelerated greedy), both in theory and practice? In this paper, we develop the first linear-time algorithm for maximizing a general monotone submodular function subject to a cardinality constraint. We show that our randomized algorithm, STOCHASTIC-GREEDY, can achieve a @math approximation guarantee, in expectation, to the optimum solution in time linear in the size of the data and independent of the cardinality constraint. We empirically demonstrate the effectiveness of our algorithm on submodular functions arising in data summarization, including training large-scale kernel methods, exemplar-based clustering, and sensor placement. We observe that STOCHASTIC-GREEDY practically achieves the same utility value as lazy greedy but runs much faster. More surprisingly, we observe that in many practical scenarios STOCHASTIC-GREEDY does not evaluate the whole fraction of data points even once and still achieves indistinguishable results compared to lazy greedy. In this work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over a down-closed polytope. The approximation guarantee is 0.372 and it is the first improvement over the 1 e approximation achieved by the unified Continuous Greedy algorithm [, FOCS 2011]. We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, @math approximation, using @math function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to @math running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.