Auction algorithms for network flow problems: A tutorial introduction

Abstract

This paper surveys a new and comprehensive class of algorithms for solving the classical linear network flow problem and its various special cases such as shortest path, max-flow, assignment, transportation, and transhipment problems. The prototype method, from which the other algorithms can be derived, is the auction algorithm for the assignment problem. This is an intuitive method that operates like a real auction where persons compete for objects by raising their prices through competitive bidding; the prices can be viewed as dual variables. Conceptually, auction algorithms represent a significant departure from the cost improvement idea that underlies primal simplex and dual ascent methods; at any one iteration, they may deteriorate both the primal and the dual cost. Auction algorithms perform very well for several important types of problems, both in theory and in practice, and they are also well suited for parallel computation.

A distributed auction algorithm for the assignment problem

“…We achieve this goal by defining a distributed market where the agents are able to bid for the task to which they wish to be assigned. Unlike prior work that assumes either a single central auctioneer [22] or multiple parallel processing units (corresponding to the agents) composed in an asynchronous shared memory system with global information [23], we propose a distributed control framework, where every agent independently and using only local information, is able to determine a task to be assigned to (Fig. 1).…”

“…As in [22], let every task j have a price p j (t) ≥ 0 at time t, which every agent i that gets assigned to it has to pay. Then, the net value of task j for agent i is β ij − p j (t) and every agent i would like to be assigned to a task j that provides it with a maximum net value…”

“…As shown in [22], designing a market that guarantees an equilibrium assignment is not straightforward, due to the possibility of cycles in the algorithm, resulting from several agents requesting to be assigned to a smaller number of equally desirable tasks without, however, raising their prices. For this, we employ the notion of an almost equilibrium assignment and set of prices, introduced in [22].…”

“…For this, we employ the notion of an almost equilibrium assignment and set of prices, introduced in [22]. This notion of an equilibrium is motivated by real auctions, where every bid for a task must raise its price by a minimum positive increment and the agents must take risks to win their preferred tasks.…”

“…The need, however, for more efficient and reliable algorithms, has recently lead to decentralized approaches, where computation and ideally also information is distributed among multiple parallel processing units. One such approach is the auction algorithm proposed in [22], [23] that based on an appropriate choice of bids, determines the prices of the tasks and renders them more or less attractive for the agents to bid for. Although, the assignment benefit does not necessarily increase monotonically, it can be shown that eventually an optimal assignment is discovered.…”