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Belief propagation: an asymptotically optimal algorithm for the random assignment problem
"... The random assignment problem concerns finding the minimum cost assignment or matching in a complete bipartite graph with edge weights being i.i.d. with some distribution, say exponential(1) distribution. In a remarkable result by Aldous (2001), it was shown that the average cost of such an assignme ..."
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The random assignment problem concerns finding the minimum cost assignment or matching in a complete bipartite graph with edge weights being i.i.d. with some distribution, say exponential(1) distribution. In a remarkable result by Aldous (2001), it was shown that the average cost of such an assignment converges to ζ(2) = π 2 /6 as the size of bipartite graph increases to ∞; thus proving conjecture of Mézard and Parisi (1987) based on replica method arising from statistical physics insights. This conjecture also suggested a heuristic for finding such an assignment, which is an instance of the wellknown heuristic Belief Propagation (BP) discussed by Pearl (1987). In a recent work by Bayati, Shah and Sharma (2005), BP was shown to find correct solution in O(n 3) time for the instance of assignment problem over graph of size n with arbitrary weights. In contrast, in this paper we establish that the BP finds an asymptotically correct assignment in O(n 2) time with high probability for the random assignment problem for a large class of edge weight distributions. Thus, BP is essentially an optimal algorithm for the assignment problem under random setup. Our result utilizes result of Aldous (2001) and the notion of local weak convergence. Key nontrivial steps in establishing our result involve proving attractiveness (aka decay of correlation) of an operator acting on space of distributions corresponding to the mincost matching on Poisson Weighted Infinite Tree (PWIT) and establishing uniform convergence of dynamics of BP on bipartite graph to an appropriately defined dynamics on PWIT. Key words: Belief propagation; random assignment problem; local weak convergence; correlation decay; Poisson weighted infinite tree.
Optimality of belief propagation for random assignment problem
 SYMPOSIUM ON DISCRETE ALGORITHMS
, 2009
"... The assignment problem concerns finding the minimumcost perfect matching in a complete weighted n × n bipartite graph. Any algorithm for this classical question clearly requires Ω(n 2) time, and the best known one (Edmonds and Karp, 1972) finds solution in O(n³). For decades, it has remained unknow ..."
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The assignment problem concerns finding the minimumcost perfect matching in a complete weighted n × n bipartite graph. Any algorithm for this classical question clearly requires Ω(n 2) time, and the best known one (Edmonds and Karp, 1972) finds solution in O(n³). For decades, it has remained unknown whether optimal computation time is closer to n 3 or n 2. We provide answer to this question for random instance of assignment problem. Specifically, we establish that Belief Propagation finds solution in O(n²) time when edgeweights are i.i.d. with light tailed distribution.
Edge Flows in the Complete RandomLengths Network
, 2007
"... Consider the complete nvertex graph whose edgelengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the n → ∞ limit we find explicitly the emp ..."
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Consider the complete nvertex graph whose edgelengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the n → ∞ limit we find explicitly the empirical distribution of these edgeflows, suitably normalized.
Efficient algorithms for threedimensional axial and planar random assignment problems
, 2013
"... Beautiful formulas are known for the expected cost of random twodimensional assignment problems, but in higher dimensions, even the scaling is not known. In three dimensions and above, the problem has natural “Axial ” and “Planar ” versions, both of which are NPhard. For 3dimensional Axial random ..."
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Beautiful formulas are known for the expected cost of random twodimensional assignment problems, but in higher dimensions, even the scaling is not known. In three dimensions and above, the problem has natural “Axial ” and “Planar ” versions, both of which are NPhard. For 3dimensional Axial random assignment instances of size n, the cost scales as Ω(1/n), and a main result of the present paper is a lineartime algorithm that, with high probability, finds a solution of cost O(n −1+o(1)). For 3dimensional Planar assignment, the lower bound is Ω(n), and we give a new efficient matchingbased algorithm that with high probability returns a solution with cost O(n log n). 1
LOCAL TAIL BOUNDS FOR FUNCTIONS OF INDEPENDENT RANDOM VARIABLES
, 2008
"... It is shown that functions defined on {0,1,...,r − 1} n satisfying certain conditions of bounded differences that guarantee subGaussian tail behavior also satisfy a much stronger “local” subGaussian property. For selfbounding and configuration functions we derive analogous locally subexponential ..."
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It is shown that functions defined on {0,1,...,r − 1} n satisfying certain conditions of bounded differences that guarantee subGaussian tail behavior also satisfy a much stronger “local” subGaussian property. For selfbounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand’s [Ann. Probab. 22 (1994) 1576–1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on {0,1,...,r − 1} n for r ≥ 2.
COMPLETING A k − 1 ASSIGNMENT
, 2004
"... Abstract. We consider the distribution of the value of the optimal kassignment in an m × nmatrix, where the entries are independent exponential random variables with arbitrary rates. We give closed formulas for both the Laplace transform of this random variable and for its expected value under the ..."
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Abstract. We consider the distribution of the value of the optimal kassignment in an m × nmatrix, where the entries are independent exponential random variables with arbitrary rates. We give closed formulas for both the Laplace transform of this random variable and for its expected value under the condition that there is a zerocost k − 1assignment. 1.
Author manuscript, published in "Mathematics of Operations Research (2009)" Belief propagation: an asymptotically optimal algorithm for the random assignment problem
, 2009
"... The random assignment problem asks for the minimumcost perfect matching in the complete n × n bipartite graph Knn with i.i.d. edge weights, say uniform on [0, 1]. In a remarkable work by Aldous (2001), the optimal cost was shown to converge to ζ(2) as n → ∞, as conjectured by Mézard and Parisi (198 ..."
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The random assignment problem asks for the minimumcost perfect matching in the complete n × n bipartite graph Knn with i.i.d. edge weights, say uniform on [0, 1]. In a remarkable work by Aldous (2001), the optimal cost was shown to converge to ζ(2) as n → ∞, as conjectured by Mézard and Parisi (1987) through the socalled cavity method. The latter also suggested a nonrigorous decentralized strategy for finding the optimum, which turned out to be an instance of the Belief Propagation (BP) heuristic discussed by Pearl (1987). In this paper we use the objective method to analyze the performance of BP as the size of the underlying graph becomes large. Specifically, we establish that the dynamic of BP on Knn converges in distribution as n → ∞ to an appropriately defined dynamic on the Poisson Weighted Infinite Tree, and we then prove correlation decay for this limiting dynamic. As a consequence, we obtain that BP finds an asymptotically correct assignment in O(n 2) time only. This contrasts with both the worstcase upper bound for convergence of BP derived by Bayati, Shah and Sharma (2005) and the bestknown computational cost of Θ(n 3) achieved by Edmonds and Karp’s algorithm (1972).