A distributed auction algorithm for the assignment problem

Abstract

Abstract-The assignment problem constitutes one of the fundamental problems in the context of linear programming. Besides its theoretical significance, its frequent appearance in the areas of distributed control and facility allocation, where the problems' size and the cost for global computation and information can be highly prohibitive, gives rise to the need for local solutions that dynamically assign distinct agents to distinct tasks, while maximizing the total assignment benefit. In this paper, we consider the linear assignment problem in the context of networked systems, where the main challenge is dealing with the lack of global information due to the limited communication capabilities of the agents. We address this challenge by means of a distributed auction algorithm, where the agents are able to bid for the task to which they wish to be assigned. The desired assignment relies on an appropriate selection of bids that determine the prices of the tasks and render them more or less attractive for the agents to bid for. Up to date pricing information, necessary for accurate bidding, can be obtained in a multi-hop fashion by means of local communication between adjacent agents. Our algorithm is an extension to the parallel auction algorithm proposed by Bertsekas et al to the case where only local information is available and it is shown to always converge to an assignment that maximizes the total assignment benefit within a linear approximation of the optimal one.

A Survey on Aerial Swarm Robotics

“…In some cases, this may cause the agents to compete against each other, while in other cases the locally optimal behavior may also (approximately) maximize the global reward. The latter is the premise of game-theoretic methods and auction algorithms [15], [16]. In auctions, for instance, maximizing the local benefit also maximizes the net global benefit (defined as the sum of individual benefits) and concurrently solves the dual pricing problem [15].…”

“…minimizing the equivalent collective cost). Parallel or distributed algorithms to solve target assignment include many variants of distributed auction algorithms [16], [25], [109]- [111] and decentralized hierarchical strategies [112] that approximate true optimality of Kuhn's centralized Hungarian method. As an illustration of the computational complexity of auction algorithms, the number of computations required for the distributed algorithm from [16] to converge is O(∆n 2 ), where ∆ is the diameter of the communication graph underlying the network of agents participating in the auction.…”

Optimal Market-based Multi-Robot Task Allocation via Strategic Pricing

“…These extensions all increase the implementation complexity significantly. Recently [26] extended AUCTION to use local information over a multi-hop network; unsurprisingly it was based on the initial version.…”

“…This can be achieved in different ways depending on the communication model involved (a typical example is via broadcast over a publisher/subscriber mechanism as used in MURDOCH [9,10]). If communication has limited range, conflicted customers can locate and communicate with each other via multi-hop message passing (see [26]); additional bookkeeping may be required if connectivity is not maintained or the topology changes during execution of the algorithm.…”

Multi-Level Partitioning and Distribution of the Assignment Problem for Large-Scale Multi-Robot Task Allocation

“…Some algebraic, greedy, market-based, and swarm-intelligence algorithms fall into this category (see e.g., [3,4,5]). Important prior work has looked at parallel implementations of known centralized algorithms (e.g.,, [6,7]) and under a variety of differing communication constraints (e.g.,, [8]). The present work attempts to combine the merits of centralized and decentralized methods: it employs global information infrequently, and even then uses that information to distribute the work in performing the task allocation.…”