We consider an optimization problem where the decision variable is a string of bounded length. For some time there has been an interest in bounding the performance of the greedy strategy for this problem. Here, we provide weakened sufficient conditions for the greedy strategy to be bounded by a factor of (1 − (1 − 1 K)K), where K is the optimization horizon length. Specifically, we introduce the notions of K-submodularity and K-GO-concavity, which together are sufficient for this bound to hold. By introducing a notion of curvature η ∈ (0; 1), we prove an even tighter bound with the factor (1 η)(1−e−η). Finally, we illustrate the strength of our results by considering two example applications. We show that our results provide weaker conditions on parameter values in these applications than in previous results.
In this section, we first introduce some definitions related to strings and curvature. We then review the main results from @cite_12 and @cite_11 . Specifically, the results there provide sufficient conditions on the objective function @math in ) such that the greedy strategy achieves a @math -bound.
@cite_12 @cite_11
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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. The problem of choosing a string of actions to optimize an objective function that is string submodular has been considered in [1]. There it is shown 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 (1-e-1)-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 σ, then the greedy strategy achieves at least a 1 σ(1 - e-σ)-approximation of the optimal strategy. If the objective function has total forward curvature c, then the greedy strategy achieves at least a (1 - e)-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. 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. 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 .