Distributed Submodular Maximization with Parallel Execution

The submodular maximization problem is widely applicable in many engineering problems where objectives exhibit diminishing returns. While this problem is known to be NP-hard for certain subclasses of objective functions, there is a greedy algorithm which guarantees approximation at least 1/2 of the optimal solution. This greedy algorithm can be implemented with a set of agents, each making a decision sequentially based on the choices of all prior agents. In this paper, we consider a generalization of the greedy algorithm in which agents can make decisions in parallel, rather than strictly in sequence. In particular, we are interested in partitioning the agents, where a set of agents in the partition all make a decision simultaneously based on the choices of prior agents, so that the algorithm terminates in limited iterations. We provide bounds on the performance of this parallelized version of the greedy algorithm and show that dividing the agents evenly among the sets in the partition yields an optimal structure. It is shown that such optimal structures holds even under very relaxed information constraints. We additionally show that this optimal structure is still near-optimal, even when additional information (i.e., total curvature) is known about the objective function

The Impact of Message Passing in Agent-Based Submodular Maximization

Submodular maximization problems are a relevant model set for many real-world applications. Since these problems are generally NP-Hard, many methods have been developed to approximate the optimal solution in polynomial time. One such approach uses an agent-based greedy algorithm, where the goal is for each agent to choose an action from its action set such that the union of all actions chosen is as high-valued as possible. Recent work has shown how the performance of the greedy algorithm degrades as the amount of information shared among the agents decreases, whereas this work addresses the scenario where agents are capable of sharing more information than allowed in the greedy algorithm. Specifically, we show how performance guarantees increase as agents are capable of passing messages, which can augment the allowable decision set for each agent. Under these circumstances, we show a near-optimal method for message passing, and how much such an algorithm could increase performance for any given problem instance

The Impact of Information in Cooperative and Noncooperative Systems

Large-scale autonomous systems are systems comprised of many components, each acting according to its own preferences, local information and capabilities. Such systems are ubiquitous in our world and include automated warehouses, UAV swarms, traffic systems, sensor networks, the internet of things, auctions, ridesharing systems, and social networks. We focus on autonomous systems which are engineered: each component is human-designed.While such systems are attractive because they can process a high amount of data and are generally robust against single points of failure, there are often many challenges in their design. System designers must take into account that each component has its own capabilities, model of the environment, data set, and local objective. Furthermore, the system designer often cannot make decisions for each component at each time step, rather, decision-making rules are assigned to allow components to react autonomously. Small adjustments to such rules can often have cascading effects throughout the system. Finally, the interconnected nature of the system opens doors to new kinds of system-wide attacks and vulnerabilities.In this work, we focus on the challenge of information sharing constraints: each agent does not have access to all of the system information at every given time step. These constraints often arise naturally (i.e., no router can access all available data on the internet before making a routing decision), but they can also arise from privacy, trust or political issues. Thus it is imperative for the system designer to understand the relevant information constraints on the system and their effects on the emergent behavior. In this work we endeavor to answer the two following questions:1. How do a set of information sharing constraints impact the resulting emergent system-wide behavior?2. How can a system designer strategically set decision-making rules for the components to offset any negative effects caused by information sharing constraints?We answer the first question by assuming that the emergent behavior is a system equilibrium, and then comparing the value of the worst-case equilibrium to the value of the optimal decision set, where value is based on the system designer objective. Different types of information sharing constraints among the components are evaluated on this basis. We answer the second question in certain settings by showing that small deviations from standard decision-making rules can improve the system's performance guarantees.These questions are addressed in two settings: first in a cooperative setting, where the system designer can design the decision-making rules for each agent. The system designer objective function is assumed to be submodular, and this property is leveraged to show closeness of equilibrium to optimal. The second is a noncooperative setting, where the system design must operate in the presence of an attacker. Here, the constraints on information sharing are related to how much knowledge about the attacker the other players have

The Impact of Information in Distributed Submodular Maximization

The maximization of submodular functions is an NP-Hard problem for certain subclasses of functions, for which a simple greedy algorithm has been shown to guarantee a solution whose quality is within 1/2 of the optimal. When this algorithm is implemented in a distributed way, agents sequentially make decisions based on the decisions of all previous agents. This work explores how limited access to the decisions of previous agents affects the quality of the solution of the greedy algorithm. Specifically, we provide tight upper and lower bounds on how well the algorithm performs, as a function of the information available to each agent. Intuitively, the results show that performance roughly degrades proportionally to the size of the largest group of agents which make decisions independently. Additionally, we consider the case where a system designer is given a set of agents and a global limit on the amount of information that can be accessed. Our results show that the best designs partition the agents into equally-sized sets and allow agents to access the decisions of all previous agents within the same set

The Impact of Information in Distributed Submodular Maximization

The maximization of submodular functions is an NP-Hard problem for certain subclasses of functions, for which a simple greedy algorithm has been shown to guarantee a solution whose quality is within 1/2 of the optimal. When this algorithm is implemented in a distributed way, agents sequentially make decisions based on the decisions of all previous agents. This work explores how limited access to the decisions of previous agents affects the quality of the solution of the greedy algorithm. Specifically, we provide tight upper and lower bounds on how well the algorithm performs, as a function of the information available to each agent. Intuitively, the results show that performance roughly degrades proportionally to the size of the largest group of agents which make decisions independently. Additionally, we consider the case where a system designer is given a set of agents and a global limit on the amount of information that can be accessed. Our results show that the best designs partition the agents into equally-sized sets and allow agents to access the decisions of all previous agents within the same set

Impact of Information in Greedy Submodular Maximization

International audience<p>The maximization of submodular functions is a well-studied topic due to its application in many common engineering problems. Because this problem has been shown to be NP-Hard for certain subclasses of functions, much work has been done to develop efficient algorithms to approximate an optimal solution. Among these is a simple greedy algorithm, which has been shown to guarantee a solution within 1/2 theoptimal. However, when this algorithm is implemented in a distributed way, it requires all agents to share information with one another - a costly constraint for some applications. This work explores how the degradation of information sharing among the agents affects the performance of the distributed greedy algorithm. For any underlying communication graph structure, we show results for how well the distributed greedy algorithm can perform. In addition, for applications where the number of agents and number of communication links is fixed, we identify near-optimal graph structures with the highest performance guarantees. This result can inform system designers as to the most impactful places to insert communication links.</p