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57
A simple proof of the Parisi and COPPERSMITHSORKIN FORMULAS FOR THE RANDOM ASSIGNMENT PROBLEM
 LINKÖPING STUDIES IN MATHEMATICS, NO. 6
, 2005
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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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Cited by 9 (0 self)
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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.
Exact formulas and limits for a class of random optimization problems
 LINKÖPING STUDIES IN MATHEMATICS
, 2005
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Maximum independent sets on random regular graphs
, 2013
"... Abstract. We determine the asymptotics of the independence number of the random dregular graph for all d ě d0. It is highly concentrated, with constantorder fluctuations around nα ‹ ´ c ‹ logn for explicit constants α‹pdq and c‹pdq. Our proof rigorously confirms the onestep replica symmetry brea ..."
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Cited by 7 (2 self)
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Abstract. We determine the asymptotics of the independence number of the random dregular graph for all d ě d0. It is highly concentrated, with constantorder fluctuations around nα ‹ ´ c ‹ logn for explicit constants α‹pdq and c‹pdq. Our proof rigorously confirms the onestep replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs. 1.
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.
A necessary and sufficient condition for the tailtriviality of a recursive tree process. Sankhyā 68
, 2006
"... Given a recursive distributional equation (RDE) and a solution µ of it, we consider the tree indexed invariant process called the recursive tree process (RTP) with marginal µ. We introduce a new type of bivariate uniqueness property which is different from the one defined by Aldous and Bandyopadhyay ..."
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Cited by 5 (3 self)
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Given a recursive distributional equation (RDE) and a solution µ of it, we consider the tree indexed invariant process called the recursive tree process (RTP) with marginal µ. We introduce a new type of bivariate uniqueness property which is different from the one defined by Aldous and Bandyopadhyay [5], and we prove that this property is equivalent to tailtriviality for the RTP, thus obtaining a necessary and sufficient condition to determine tailtriviality for a RTP in general. As an application we consider Aldous ’ construction of the frozen percolation process on a infinite regular tree [3] and show that the associated RTP has a trivial tail. AMS 2000 subject classification: 60K35, 60G10, 60G20. Key words and phrases: Bivariate uniqueness, distributional identities, endogeny,
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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Cited by 4 (1 self)
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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.
The Limit in the Mean Field Bipartite Travelling Salesman Problem
, 2006
"... The edges of the complete bipartite graph Kn,n are assigned independent lengths from uniform distribution on the interval [0,1]. Let Ln be the length of the minimum travelling salesman tour. We prove that as n tends to infinity, Ln converges in probability to a certain number, approximately 4.0831. ..."
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The edges of the complete bipartite graph Kn,n are assigned independent lengths from uniform distribution on the interval [0,1]. Let Ln be the length of the minimum travelling salesman tour. We prove that as n tends to infinity, Ln converges in probability to a certain number, approximately 4.0831. This number is characterized as the area of the region in the xyplane. x,y ≥ 0, (1 + x/2) · e −x + (1 + y/2) · e −y ≥ 1 1
ADDENDUM TO “THE MINIMAL SPANNING TREE IN A COMPLETE GRAPH AND A FUNCTIONAL LIMIT THEOREM FOR TREES IN A RANDOM GRAPH”
, 2006
"... In the article “The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph” by Janson [6] it is shown that the minimal weight Wn of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges has an asymptot ..."
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Cited by 4 (1 self)
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In the article “The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph” by Janson [6] it is shown that the minimal weight Wn of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges has an asymptotic normal distribution. The same holds with exponentially distributed weights with mean 1. The mean converges, as shown in the classical paper by A. Frieze [3], to ζ(3), and the asymptotic variance is σ 2 /n for a positive constant σ 2; more precisely, see [6, Theorem 1], n 1/2 � Wn − ζ(3) � d − → N(0, σ 2) as n → ∞. The constant σ2 was given in [6] by the complicated expression σ 2 = π4 ∞ � ∞ � ∞ � (i + k − 1)! k − 2 45 k (i + j) i−2j ≈ 1.6857. (1) i! k! (i + j + k) i+k+2 i=0 j=1 k=1 This expression has now been evaluated by Wästlund [7], who found the simple result σ 2 = 6ζ(4) − 4ζ(3). (2) For the proof, see [7]. The proof of Lemma 1 there may be simplified since (3) was shown by N. H. Abel [1]; this has also been noted by Piet Van Mieghem in a personal communication. In principle, (2) lies within the scope of automated summation techniques.