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.
Conforti and Cornu ' e jols @cite_18 defined the @math of non-decreasing submodular function @math as They showed that greedy algorithm has an approximation ratio of @math for the problem of maximizing a nondecreasing submodular function with curvature at most @math subject to a single matroid constraint. In the special case of a uniform matroid, they were able to show that the greedy is a @math -approximation algorithm. Later, Vondr ' a k @cite_15 considered the continuous greedy algorithm in the setting of bounded curvature. He introduced the notion of , which is a weaker notion than total curvature, and showed that the continuous greedy algorithm is a @math -approximation for maximizing a nondecreasing submodular function @math subject to an arbitrary matroid constraint whenever @math has curvature at most @math with respect to the optimum. He also showed that it is impossible to obtain a @math -approximation in this setting when evaluating @math on only a polynomial number of sets. Unfortunately, unlike total curvature, it is in general not possible to compute the curvature of a function with respect to the optimum, as it requires knowledge of an optimal solution.
@cite_18 @cite_15
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Let (X;I) be a matroid and let f : 2 X ! R+ be a monotone submodular function. The curvature of f is the minimum c 2 [0; 1] such that for any S X and j 2 X n S, f (S [ fjg) f (S) (1 c)f (fjg). We consider the optimization problem maxff (S) : S 2 Ig. It is known that the greedy algorithm yields a 1=2-approximation for this problem [10], and 1 1+c -approximation when f has curvature c [3]. For the uniform matroid, it was known that the greedy algorithm yields an improved 1 (1 e c )-approximation [3]. In this paper, we analyze the continuous greedy algorithm [19] and prove that it gives a 1 c (1 e c )-approximation for any matroid. Moreover, we show that this holds for a relaxed notion of curvature, curvature with respect to the optimum, and we prove that any better approximation under these conditions would require an exponential number of value queries. Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a @math -approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. If the objective function has total forward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint. We investigate two applications of string submodular functions with curvature constraints: 1) choosing a string of actions to maximize the expected fraction of accomplished tasks; and 2) designing a string of measurement matrices such that the information gain is maximized. Consider a suboptimal solution S for a maximization problem. Suppose S's value is small compared to an optimal solution OPT to the problem, yet S is structurally similar to OPT. A natural question in this setting is whether there is a way of improving S based solely on this information. In this paper we introduce the Structural Continuous Greedy Algorithm, answering this question affirmatively in the setting of the Nonmonotone Submodular Maximization Problem. We improve on the best approximation factor known for this problem. In the Nonmonotone Submodular Maximization Problem we are given a non-negative submodular function f, and the objective is to find a subset maximizing f. Our method yields an 0.42-approximation for this problem, improving on the current best approximation factor of 0.41 given by Gharan and Vondrak [5]. On the other hand, [4] showed a lower bound of 0.5 for this problem. Abstract For the problem maxlcub; Z ( S ): S is an independent set in the matroid X rcub;, it is well-known that the greedy algorithm finds an optimal solution when Z is an additive set function (Rado-Edmonds theorem). Fisher, Nemhauser and Wolsey have shown that, when Z is a nondecreasing submodular set function satisfying Z (∅)=0, the greedy algorithm finds a solution with value at least half the optimum value. In this paper we show that it finds a solution with value at least 1 (1 + α) times the optimum value, where α is a parameter which represents the ‘total curvature’ of Z . This parameter satisfies 0≤α≤1 and α=0 if and only if the set function Z is additive. Thus the theorems of Rado-Edmonds and Fisher-Nemhauser-Wolsey are both contained in the bound 1 (1 + α). We show that this bound is best possible in terms of α. Another bound which generalizes the Rado-Edmonds theorem is given in terms of a ‘greedy curvature’ of the set function. Unlike the first bound, this bound can prove the optimality of the greedy algorithm even in instances where Z is not additive. A third bound, in terms of the rank and the girth of X , unifies and generalizes the bounds (e−1) e known for uniform matroids and 1 2 for general matroids. We also analyze the performance of the greedy algorithm when X is an independence system instead of a matroid. Then we derive two bounds, both tight: The first one is [1−(1−α K ) k ] α where K and k are the sizes of the largest and smallest maximal independent sets in X respectively; the second one is 1 ( p +α) where p is the minimum number of matroids that must be intersected to obtain X .