  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.
