
Contents
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16.1 The assignment problem and random assignment ensembles 16.1 The assignment problem and random assignment ensembles
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16.2 Message passing and its probabilistic analysis 16.2 Message passing and its probabilistic analysis
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16.2.1 Statistical-physics formulation and counting 16.2.1 Statistical-physics formulation and counting
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16.2.2 The belief propagation equations 16.2.2 The belief propagation equations
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16.2.3 Zero temperature: The min-sum algorithm 16.2.3 Zero temperature: The min-sum algorithm
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16.2.4 The distributional fixed point and ζ(2) 16.2.4 The distributional fixed point and ζ(2)
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16.2.5 Non-zero temperature and stability analysis 16.2.5 Non-zero temperature and stability analysis
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16.3 A polynomial message-passing algorithm 16.3 A polynomial message-passing algorithm
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16.3.1 The computation tree 16.3.1 The computation tree
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16.3.2 Proof of convergence of the min-sum algorithm 16.3.2 Proof of convergence of the min-sum algorithm
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16.3.3 A few remarks 16.3.3 A few remarks
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16.4 Combinatorial results 16.4 Combinatorial results
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16.4.1 The Coppersmith–Sorkin and Parisi formulae 16.4.1 The Coppersmith–Sorkin and Parisi formulae
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16.4.2 From k-assignment to k+1-assignment 16.4.2 From k-assignment to k+1-assignment
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16.4.3 Proof of Theorem 16.4 16.4.3 Proof of Theorem 16.4
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16.5 An exercise: Multi-index assignment 16.5 An exercise: Multi-index assignment
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Notes Notes
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Cite
Abstract
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.
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