We consider the classical linear assignment problem, and we introduce new auction algorithms for its optimal and suboptimal solution. The algorithms are founded on duality theory, and are related to ideas of competitive bidding by persons for objects and the attendant market equilibrium, which underlie real-life auction processes. We distinguish between two fundamentally different types of bidding mechanisms: aggressive and cooperative. Mathematically, aggressive bidding relies on a notion of approximate coordinate descent in dual space, an epsilon-complementary slackness condition to regulate the amount of descent approximation, and the idea of epsilon-scaling to resolve efficiently the price wars that occur naturally as multiple bidders compete for a smaller number of valuable objects. Cooperative bidding avoids price wars through detection and cooperative resolution of any competitive impasse that involves a group of persons. We discuss the relations between the aggressive and the cooperative bidding approaches, we derive new algorithms and variations that combine ideas from both of them, and we also make connections with other primal-dual methods, including the Hungarian method. Furthermore, our discussion points the way to algorithmic extensions that apply more broadly to network optimization, including shortest path, max-flow, transportation, and minimum cost flow problems with both linear and convex cost functions.

翻译：本文考虑经典线性赋值问题，并引入用于求解其最优解和次优解的新拍卖算法。这些算法基于对偶理论，并与个人竞标物品及其伴随的市场均衡思想相关，这些思想构成了现实拍卖过程的基础。我们区分了两种根本不同类型的竞标机制：激进型与合作型。从数学角度看，激进竞价依赖于对偶空间中的近似坐标下降概念、用于调节下降近似程度的ε互补松弛条件，以及通过ε缩放高效解决多个竞标者争夺少量高价值物品时自然出现的价格战问题。合作竞价则通过检测并合作解决涉及一群人的任何竞争僵局来避免价格战。我们讨论了激进与合作竞价方法之间的关系，推导了结合两者思想的新算法及其变体，并与包括匈牙利算法在内的其他原始-对偶方法建立了联系。此外，我们的讨论为算法扩展指明了方向，使其更广泛地适用于网络优化问题，包括具有线性及凸成本函数的最短路径、最大流、运输和最小费用流问题。