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51
Percolationlike scaling exponents for minimal paths and trees in the stochastic meanfield
, 2005
"... Abstract In the mean field (or random link) model there are n points andinterpoint distances are independent random variables. For 0 < ` < 1and in the n! 1 limit, let ffi(`) = 1/n * (maximum number of stepsin a path whose average steplength is <= `). The function ffi(`) isanalogous to th ..."
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Abstract In the mean field (or random link) model there are n points andinterpoint distances are independent random variables. For 0 < ` < 1and in the n! 1 limit, let ffi(`) = 1/n * (maximum number of stepsin a path whose average steplength is <= `). The function ffi(`) isanalogous to the percolation function in percolation theory: there is a critical value ` * = e1 at which ffi(*) becomes nonzero, and (presumably) a scaling exponent fi in the sense ffi(`) i ( ` `*)fi. Recentlydeveloped probabilistic methodology (in some sense a rephrasing of the cavity method developed in the 1980s by M'ezard and Parisi) provides a simple albeit nonrigorous way of writing down such functions in terms of solutions of fixedpoint equations for probability distributions. Solving numerically gives convincing evidence that fi = 3. Aparallel study with trees and connected edgesets in place of paths gives scaling exponent 2, while the analog for classical percolation has scaling exponent 1. The new exponents coincide with those recently found in a different context (comparing optimal and nearoptimal solutions ofthe meanfield TSP and MST problems), and reinforce the suggestion that scaling exponents determine universality classes for optimizationproblems on random points. Key words and phrases. Combinatorial optimization, mean field model, percolation, probabilistic analysis of algorithms, scaling exponent,
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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Cited by 9 (4 self)
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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.
On the Random 2Stage Minimum Spanning Tree
, 2004
"... It is known [6] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to (3) = . Here we consider the following stochastic twostage version of this op ..."
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Cited by 9 (3 self)
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It is known [6] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to (3) = . Here we consider the following stochastic twostage version of this optimization problem. There are two sets of edge costs c M : E ! R and c T : E ! R , called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs c M (e) and c T (e) are independent random variables, uniformly distributed in [0; 1]. The Monday costs are revealed rst. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c M (e), or to wait until its Tuesday price c T (e) appears. The set of edges XM bought on Monday is then completed by the set of edges X T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost (3)=2 + o(1). We show that in the case of twostage optimization, the expected value of the optimal cost exceeds (3)=2 by an absolute constant > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than and completes them on Tuesday in an optimal way, and show that the optimal choice for is = 1=n with the expected cost (3) 1=2 + o(1). The threshold heuristic is shown to be suboptimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning outarborescence rooted at a xed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 1=e + o(1).
Replica symmetry of the minimum matching
, 2011
"... We establish the soundness of the replica symmetric ansatz introduced by M. Mézard and G. Parisi for the minimum matching problem in the pseudodimension d mean field model for d ≥ 1. The case d = 1 corresponds to the π 2 /6limit for the assignment problem proved by D. Aldous in 2001. We introduce ..."
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We establish the soundness of the replica symmetric ansatz introduced by M. Mézard and G. Parisi for the minimum matching problem in the pseudodimension d mean field model for d ≥ 1. The case d = 1 corresponds to the π 2 /6limit for the assignment problem proved by D. Aldous in 2001. We introduce a gametheoretical framework by which we establish the analogous limit also for d> 1. 1
Exact formulas and limits for a class of random optimization problems
 LINKÖPING STUDIES IN MATHEMATICS
, 2005
"... ..."
RANDOM MATCHING PROBLEMS ON THE Complete Graph
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 2008
"... The edges of the complete graph on n vertices are assigned independent exponentially distributed costs. A kmatching is a set of k edges of which no two have a vertex in common. We obtain explicit bounds on the expected value of the minimum total cost Ck,n of a kmatching. In particular we prove tha ..."
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Cited by 6 (4 self)
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The edges of the complete graph on n vertices are assigned independent exponentially distributed costs. A kmatching is a set of k edges of which no two have a vertex in common. We obtain explicit bounds on the expected value of the minimum total cost Ck,n of a kmatching. In particular we prove that if n = 2k then π 2 /12 < ECk,n < π 2 /12 + log n/n.
Asymptotic Results for Random Multidimensional Assignment Problems
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
, 2003
"... The multidimensional assignment problem (MAP) is a NPhard combinatorial optimization problem occurring in applications such as data association and target tracking. In this paper, we investigate characteristics of the mean optimal solution values for random MAPs with axial constraints. Througho ..."
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Cited by 6 (1 self)
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The multidimensional assignment problem (MAP) is a NPhard combinatorial optimization problem occurring in applications such as data association and target tracking. In this paper, we investigate characteristics of the mean optimal solution values for random MAPs with axial constraints. Throughout the study, we consider cost coefficients taken from three different random distributions: uniform, exponential and standard normal. In the cases of uniform and exponential costs, experimental data indicates that the mean optimal value converges to zero when the problem size increases. We give a short proof of this result for the case of exponentially distributed costs when the number of elements in each dimension is restricted to two. In the case of standard normal costs, experimental data indicates the mean optimal value goes to negative infinity with increasing problem size. Using curve fitting techniques, we develop numerical estimates of the mean optimal value for various sized problems. The experiments indicate that numerical estimates are quite accurate in predicting the optimal solution value of a random instance of the MAP.
Bivariate Uniqueness in the Logistic Recursive Distributional Equation
, 2002
"... In this work we prove the bivariate uniqueness property of the Logistic fixedpoint equation, which arise in the study of the random assignment problem, as discussed by Aldous [4]. Using this and the general framework of Aldous and Bandyopadhyay [2], we then conclude that the associated recursive tr ..."
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Cited by 4 (1 self)
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In this work we prove the bivariate uniqueness property of the Logistic fixedpoint equation, which arise in the study of the random assignment problem, as discussed by Aldous [4]. Using this and the general framework of Aldous and Bandyopadhyay [2], we then conclude that the associated recursive tree process is endogenous, and hence the Logistic variables defined in Aldous' work [4] are measurable with respect to the #field generated by the edge weights. The method involves construction of an explicit recursion to show the uniqueness of the associated integral equation. Key words and phrases. Bivariate uniqueness, distributional identity, fixedpoint equation, Logistic distribution, measurability, nonlinear integral equation, Poisson weighted infinite tree, random assignment problem. 1
Edge cover and polymatroid flow problems
, 2010
"... In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This socalled minimum edge cover problem is a relaxation of perfect matching. We show that the large n l ..."
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In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This socalled minimum edge cover problem is a relaxation of perfect matching. We show that the large n limit cost of the minimum edge cover is W (1) 2 +2W (1) ≈ 1.456, where W is the Lambert Wfunction. In particular this means that the minimum edge cover is essentially cheaper than the minimum perfect matching, whose limit cost is π 2 /6 ≈ 1.645. We obtain this result through a generalization of the perfect matching problem to a setting where we impose a (poly)matroid structure on the two vertexsets of the graph, and ask for an edge set of prescribed size connecting independent sets.