A new performance bound for submodular maximization problems and its application to multi-agent optimal coverage problems

Abstract

Several important problems in multi-agent systems, machine learning, data mining, scheduling and others, may be formulated as set function maximization problems subject to cardinality constraints. In such problems, the set (objective) functions of interest often have monotonicity and submodularity properties. Hence, the class of monotone submodular set function maximization problems has been widely studied in the literature. Owing to its challenging nature, almost all existing solutions for this class of problems are based on greedy algorithms. A seminal work on this topic has exploited the submodularity property to prove a (1-1/e) performance bound for such greedy solutions. More recent literature on this topic has been focused on exploiting different curvature properties to establish improved (tighter) performance bounds. However, such improvements come at the cost of enforcing additional assumptions and increasing computational complexity while facing significant inherent limitations. In this paper, first, a brief review of existing performance bounds is provided. Then, a new performance bound that does not require any additional assumptions and is both practical and computationally inexpensive is proposed. In particular, this new performance bound is established based on a series of upper bounds derived for the objective function that can be computed in parallel with the execution of the greedy algorithm. Finally, to highlight the effectiveness of the proposed performance bound, extensive numerical results obtained from a well-known class of multi-agent coverage problems are provided.

Introduction

The salient feature that characterizes a submodular set function is its diminishing returns property. Simply, this property means that the marginal gain (return) of adding an element to a set decreases as the set grows (accumulates new elements). In this sense, submodularity bears a similarity to concavity. However, it has been established that maximizing submodular set functions is NP-hard (Corneuejols et al., 1977, Nemhauser et al., 1978) while minimizing them can be achieved in polynomial time (Grötschel et al., 1981, Schrijver, 2000). Therefore, submodularity also has a resemblance to convexity. Despite this duality, submodular set functions appear naturally in many real-world problems such as in the coverage control (Sun et al., 2019, Sun et al., 2020), persistent monitoring (Rezazadeh & Kia, 2019), feature selection (Das & Kempe, 2008), document summarization (Lin & Bilmes, 2011), image segmentation (Jegelka & Bilmes, 2011), marketing (Kempe, Kleinberg, & Tardos, 2003), data mining (Mirzasoleiman, Karbasi, Sarkar, & Krause, 2013), machine scheduling (Liu, 2020) and recommender systems (El-Arini & Guestrin, 2011).

Motivated by its applicability, submodular maximization problems have been theoretically studied in the literature under a diverse set of conditions (Liu, Chong, Pezeshki, & Zhang, 2020). For example, the submodular objective function is assumed to be: monotone in Wang, Moran, Wang, and Pan (2016), non-monotone in Fahrbach, Mirrokni, and Zadimoghaddam (2019) and weakly submodular in Khanna, Elenberg, Dimakis, Negahban, and Ghosh (2017). Similarly, different types of set variable constraints such as cardinality (Nemhauser et al., 1978), matroid (Fisher, Nemhauser, & Wolsey, 1978), knapsack (Wolsey, 1982) and matchoid (Badanidiyuru, Karbasi, Kazemi, & Vondrak, 2020) have been considered throughout the literature. Even though submodular maximization problems have been studied under various conditions, their solutions are predominantly based on greedy algorithms. In this paper, similar to Conforti and Cornuéjols, 1984, Liu et al., 2018, Nemhauser et al., 1978 and Wang et al. (2016), we consider the class of submodular maximization problems where the objective function is monotone, the set variable is cardinality constrained, and the solution is obtained by a vanilla greedy algorithm. Such maximization problems arise naturally from applications like coverage control (Sun et al., 2019, Sun et al., 2020), persistent monitoring (Rezazadeh & Kia, 2019), machine scheduling (Liu, 2020) and resource allocation (Liu et al., 2020). However, in contrast to the prior work in Conforti and Cornuéjols, 1984, Liu et al., 2018, Nemhauser et al., 1978 and Wang et al. (2016), here we propose a novel performance bound for the obtained greedy solutions.

Formally, a performance bound of a greedy solution is a lower bound to the ratio $f^G/f^{\ast }$ so that $\beta \le f^G/f^{\ast }$, where $f^G$ and $f^{\ast }$ correspond to the objective function values under the greedy solution and the global optimal solution, respectively. For monotone submodular objective functions, the seminal papers Fisher et al. (1978) and Nemhauser et al. (1978) respectively show that $\beta =\frac{1}{2}$ when the set variable is constrained over a general matroid and $\beta =(1-{(1-\frac{1}{N})}^N)$ when the set variable’s cardinality is constrained by $N$. Note that having a performance bound closer to $1$ is preferred, as it implies that the greedy solution is almost globally optimal.

Recent work on this class of problems has shown an increasing interest in improving the aforementioned conventional performance bounds by exploiting structural properties of the underlying problem. The typical approach is first to define a curvature measure that characterizes the structural properties of the underlying objective function, the feasible space and the generated greedy solution. Then, based on this curvature measure, an improved (closer to $1$ compared to conventional counterparts) performance bound is established. For example, Conforti and Cornuéjols (1984) defined a curvature measure named total curvature based on the nature of the objective function and the feasible space. Then, a provably improved performance bound was developed using the said total curvature measure. The authors of Conforti and Cornuéjols (1984) also proposed another curvature metric named greedy curvature based on the generated greedy solution and used it to develop another performance bound. The same procedure was followed in Liu et al. (2018) and Wang et al. (2016) to propose two new curvature metrics named elemental curvature and partial curvature respectively and then to develop corresponding performance bounds.

In this work, we first review the aforementioned total, greedy, elemental and partial curvature measures proposed in Conforti and Cornuéjols, 1984, Liu et al., 2018 and Wang et al. (2016) while outlining their strengths and weaknesses. In particular, we point out that some of these curvature measures can be: (i) computationally expensive to obtain, (ii) require enforcing additional assumptions and (iii) may have inherent limitations that prevent them from providing improved performance bounds (e.g., if the submodularity property of the objective function is strong or weak). We next propose a novel curvature measure (which we named the extended greedy curvature) along with a corresponding performance bound that can be computed efficiently in parallel with the execution of the greedy algorithm. We show that this performance bound may be improved by executing extra greedy iterations (hence the name “extended”). This new performance bound does not require any additional assumptions, and it also does not suffer from the said inherent limitations of its predecessors. Finally, we use a widely studied class of multi-agent coverage problems (Sun et al., 2019, Sun et al., 2020) and implement all the aforementioned performance bounds to highlight the effectiveness of the proposed performance bound in this paper.

The paper is organized as follows. The used preliminary concepts and notations are introduced in Section 2. A brief review of existing performance bounds is provided in Section 3. Section 4 presents the details of the proposed new performance bound. The multi-agent coverage problem setup and the observed numerical results are reported in Section 5 before concluding the paper in Section 6.