text: 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.
  @cite_29: 
  @cite_34: A real-valued function z whose domain is all of the subsets of N = 1,..., n is said to be submodular if zS + zT ≥ zS ∪ T + zS ∩ T, ∀S, T ⊆ N, and nondecreasing if zS ≤ zT, ∀S ⊂ T ⊆ N. We consider the problem maxS⊂N zS: |S| ≤ K, z submodular and nondecreasing, zO = 0 . Many combinatorial optimization problems can be posed in this framework. For example, a well-known location problem and the maximization of certain boolean polynomials are in this class. We present a family of algorithms that involve the partial enumeration of all sets of cardinality q and then a greedy selection of the remaining elements, q = 0,..., K-1. For fixed K, the qth member of this family requires Onq+1 computations and is guaranteed to achieve at least @math Our main result is that this is the best performance guarantee that can be obtained by any algorithm whose number of computations does not exceed Onq+1.
  @cite_21: Given a collection F of subsets of S = 1,…, n , setcover is the problem of selecting as few as possiblesubsets from F such that their union covers S, , and maxk-cover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems areNP-hard. We prove that (1 - o (1)) ln n is a threshold below which setcover cannot be approximated efficiently, unless NP has slightlysuperpolynomial time algorithms. This closes the gap (up to low-orderterms) between the ratio of approximation achievable by the greedyalogorithm (which is (1 - o (1)) lnn), and provious results of Lund and Yanakakis, that showed hardness ofapproximation within a ratio of log 2 n 2s0.72 ln n . For max k -cover, we show an approximationthreshold of (1 - 1 e )(up tolow-order terms), under assumption that P≠NP .
  @cite_2: A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits a certain "symmetry gap" in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1-1 e)-approximation for monotone submodular maximization under a cardinality constraint, and the impossibility of (1 2+epsilon)-approximation for unconstrained (non-monotone) submodular maximization. It follows from our result that (1 2+epsilon)-approximation is also impossible for non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309-approximation for this problem, improving the previously known factor of 1 4-o(1).As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A (1 6-o(1))-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k≫=2, there is a class of matroids of fractional base packing number nu = k (k-1), such that any algorithm achieving a better than (1-1 nu)-approximation for this class would require exponentially many value queries. On the positive side, we present a 1 2 (1-1 nu-o(1))-approximation algorithm for the same problem. Our hardness results hold in fact for very special symmetric instances. For such symmetric instances, we show that the approximation factors of 1 2 (for submodular maximization subject to a matroid constraint) and 1-1 nu (for a matroid base constraint) can be achieved algorithmically and hence are optimal.
summary: 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 .