Max-product "belief propagation" is an iterative, local, message-passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding, computer vision and combinatorial optimization which involve graphs with many cycles, theoretical results about both correctness and convergence of the algorithm are known in few cases [21], [18], [23], [16].In this paper we consider the problem of finding the Maximum Weight Matching (MWM) in a weighted complete bipartite graph. We define a probability distribution on the bipartite graph whose MAP assignment corresponds to the MWM. We use the max-product algorithm for finding the MAP of this distribution or equivalently, the MWM on the bipartite graph. Even though the underlying bipartite graph has many short cycles, we find that surprisingly, the max-product algorithm always converges to the correct MAP assignment as long as the MAP assignment is unique. We provide a bound on the number of iterations required by the algorithm and evaluate the computational cost of the algorithm. We find that for a graph of size n, the computational cost of the algorithm scales as O(n 3 ), which is the same as the computational cost of the best known algorithm. Finally, we establish the precise relation between the max-product algorithm and the celebrated auction algorithm proposed by Bertsekas. This suggests possible connections between dual algorithm and max-product algorithm for discrete optimization problems.
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Abstract-The max-product "belief propagation" algorithm is an iterative, local, message passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding and computer vision which involve graphs with many cycles, theoretical convergence results are only known for graphs which are tree-like or have a single cycle.In this paper, we consider a weighted complete bipartite graph and define a probability distribution on it whose MAP assignment corresponds to the maximum weight matching (MWM) in that graph. We analyze the fixed points of the max-product algorithm when run on this graph and prove the surprising result that even though the underlying graph has many short cycles, the maxproduct assignment converges to the correct MAP assignment. We also provide a bound on the number of iterations required by the algorithm.
Suppose that there are n jobs and n machines and it costs cij to execute job i on machine j. The assignment problem concerns the determination of a one‐to‐one assignment of jobs onto machines so as to minimize the cost of executing all the jobs. When the cij are independent and identically distributed exponentials of mean 1, Parisi [Technical Report cond‐mat/9801176, xxx LANL Archive, 1998] made the beautiful conjecture that the expected cost of the minimum assignment equals $\sum_{i=1}^n (1/i^2)$. Coppersmith and Sorkin [Random Structures Algorithms 15 (1999), 113–144] generalized Parisi's conjecture to the average value of the smallest k‐assignment when there are n jobs and m machines. Building on the previous work of Sharma and Prabhakar [Proc 40th Annu Allerton Conf Communication Control and Computing, 2002, 657–666] and Nair [Proc 40th Annu Allerton Conf Communication Control and Computing, 2002, 667–673], we resolve the Parisi and Coppersmith‐Sorkin conjectures. In the process we obtain a number of combinatorial results which may be of general interest.© 2005 Wiley Periodicals, Inc. Random Struct. Alg. 2005
Lost sales inventory models with large lead times, which arise in many practical settings, are notoriously difficult to optimize due to the curse of dimensionality. In this paper we show that when lead times are large, a very simple constant-order policy, first studied by Reiman [39], performs nearly optimally. The main insight of our work is that when the lead time is very large, such a significant amount of randomness is injected into the system between when an order for more inventory is placed and when the order is received, that "being smart" algorithmically provides almost no benefit. Our main proof technique combines a novel coupling for suprema of random walks with arguments from queueing theory.
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