# The assignment problem

# The assignment problem

This chapter discusses the use of message passing techniques in a combinatorial optimization problem assignment. Given *N* ‘agents’ and *N* ‘jobs’, and the cost matrix E(i,j) for having job *i* executed by agent *j*, the problem is to find the lowest cost assignment of jobs to agents. On the algorithmic side, the Min-Sum variant of Belief Propagation is shown to converge to an optimal solution in polynomial time. On the probabilistic side, the large *N* limit of random instances, when the costs E(i,j) are independent uniformly random variables, is studied analytically. The cost of the optimal assignment is first computed heuristically within the replica symmetric cavity method, giving the celebrated zeta(2) result. This study is confirmed by a rigorous combinatorial argument which provides a proof of the Parisi and Coppersmith–Sorkin conjectures.

*Keywords:*
assignment, minimum cost, min-sum, belief propagation, replica symmetric, cavity method, zeta(2), Coppersmith–Sorkin conjectures

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