String Submodular Functions With Curvature Constraints

Abstract

Abstract-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 ǫ, then the greedy strategy achieves at least a (1 − ǫ)-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.

Subspace Selection for Projection Maximization With Matroid Constraints

“…In this case, lower bound for the greedy algorithm is derived as a function of the elemental curvature. In [36] and [37], Zhang et al generalized the notions of total curvature and elemental curvature to string submodular functions where the objective function value depends on the order of the elements in the set. This framework is further extended to approximate dynamic programming problems by Liu et al in [38].…”

Bounds for approximate dynamic programming based on string optimization and curvature

“…In this paper, we will develop a systematic approach to deriving guaranteed bounds for ADP schemes. Our approach is inspired by our recent results in [22] and [23] on bounding the performance of greedy strategies in optimization of string-submodular functions.…”

“…To compare the greedy and optimal strategies for functions defined over strings, in [22] and [23], we have introduced the notion of string submodularity, which builds on the notion of set submodularity in combinatorial optimization. We have shown that, under string submodularity, any greedy strategy is suboptimal by a factor of at worst (1 − e −1 ), entirely consistent with the result of Nemhauser et al [13].…”

“…In this paper, inspired by the bounding techniques in [22] and [23], we develop the first systematic approach to deriving performance bounds for general ADP methods for optimal control problems. To set up our approach, in Section II, we review the relevant results from [22] and [23].…”

“…To set up our approach, in Section II, we review the relevant results from [22] and [23]. In Section III, we first describe a general optimal control problem and a class of ADP schemes for approximating optimal control solutions.…”