We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a nondecreasing submodular function or minimize a nonincreasing supermodular function in the setting of bounded total curvature @math . In the case of submodular maximization with curvature @math , we obtain a @math -approximation --- the first improvement over the greedy @math -approximation of Conforti and Cornuejols from 1984, which holds for a cardinality constraint, as well as recent approaches that hold for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and non-oblivious local search, and allows us to approximately maximize the sum of a nonnegative, nondecreasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show @math -approximation for maximization and @math -approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem.
On the negative side, Nemhauser and Wolsey @cite_34 showed that it is impossible to improve upon the bound of @math in the value oracle model, even under a single cardinality constraint. In this model, @math is given as a value oracle and an algorithm can evaluate @math on only a polynomial number of sets. Feige @cite_21 showeds that @math is the best possible approximation even when the function is given explicitly, unless @math . In later work, Vondr ' a k @cite_2 introduced the notion of the of a submodular function @math , which unifies many inapproximability results in the value oracle model, and proved new inapproximability results for some specific constrained settings. Later, Dobzinski and Vondr 'ak @cite_29 showed how these inapproximability bounds may be converted to matching complexity-theoretic bounds, which hold when @math is given explicitly, under the assumption that @math .
@cite_29 @cite_34 @cite_21 @cite_2
1997783781 2143996311 2089729462
Let @math be a monotone submodular set function, and let @math be a matroid. We consider the problem @math . It is known that the greedy algorithm yields a @math -approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73-87] for this problem. For certain special cases, e.g., @math , the greedy algorithm yields a @math -approximation. It is known that this is optimal both in the value oracle model (where the only access to @math is through a black box returning @math for a given set @math ) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177-188] and for explicitly posed instances assuming @math [U. Feige, J. ACM, 45 (1998), pp. 634-652]. In this paper, we provide a randomized @math -approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307-328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrak, Proceedings of the @math th ACM Symposium on Theory of Computing, 2008, pp. 67-74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires @math to be exponential in the original problem size, we are able to achieve a @math -approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constra We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products @math based on a set of customer samples @math . We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor @math for some constant @math , unless @math , thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within @math , @math being an upper bound on the number of nonzero values per customer, and @math under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming s Let @math be a finite lattice and let @math be a positive integer. A function @math is said to be submodular if @math for all @math . In this paper we study submodular functions when @math is a diamond. Given oracle access to @math we are interested in finding @math such that @math as efficiently as possible. We establish a min--max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular @math and an integer @math such that @math , there is a proof of this fact which can be verified in time polynomial in @math and @math ; and a pseudo-polynomial time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular @math one can find @math in time bounded by a polynomial in @math and @math . We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math .