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20
Exact formulas and limits for a class of random optimization problems
 LINKÖPING STUDIES IN MATHEMATICS
, 2005
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A heuristic for the stacker crane problem on trees which is almost surely exact
 Journal of Algorithms
, 2006
"... Abstract Given an edgeweighted transportation network G and a list of transportation requests L, the Stacker Crane Problem is to find a minimumcost tour for a server along the edges of G that serves all requests. The server has capacity one, and starts and stops at the same vertex. In this paper ..."
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Abstract Given an edgeweighted transportation network G and a list of transportation requests L, the Stacker Crane Problem is to find a minimumcost tour for a server along the edges of G that serves all requests. The server has capacity one, and starts and stops at the same vertex. In this paper, we consider the case that the transportation network G is a tree, and that the requests are chosen randomly according to a certain class of probability distributions. We show that a polynomial time algorithm by Frederickson and Guan [9], which guarantees a 4/3approximation in the worst case, on almost all inputs finds a minimumcost tour, along with a certificate of the optimality of its output. 1
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
Replica Symmetry and Combinatorial Optimization
, 2009
"... We establish the soundness of the replica symmetric ansatz (introduced by M. Mézard and G. Parisi) for minimum matching and the traveling salesman problem in the pseudodimension d mean field model for d ≥ 1. The case d = 1 of minimum matching corresponds to the π 2 /6limit for the assignment probl ..."
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We establish the soundness of the replica symmetric ansatz (introduced by M. Mézard and G. Parisi) for minimum matching and the traveling salesman problem in the pseudodimension d mean field model for d ≥ 1. The case d = 1 of minimum matching corresponds to the π 2 /6limit for the assignment problem established by D. Aldous in 2001, and the analogous limit for the d = 1 case of TSP was recently established by the author with a different method. We introduce a gametheoretical framework by which we prove the correctness of the replicacavity prediction of the corresponding limits also for d> 1.
Mean field matching and traveling salesman problems in pseudodimension 1
, 2012
"... Recent work on optimization problems in random link models has verified several conjectures originating in statistical physics and the replica and cavity methods. In particular the numerical value 2.0415 for the limit length of a traveling salesman tour in a complete graph with uniform [0, 1] edgel ..."
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Recent work on optimization problems in random link models has verified several conjectures originating in statistical physics and the replica and cavity methods. In particular the numerical value 2.0415 for the limit length of a traveling salesman tour in a complete graph with uniform [0, 1] edgelengths has been established. In this paper we show that the crucial integral equation obtained with the cavity method has a unique solution, and that the limit ground state energy obtained from this solution agrees with the rigorously derived value. Moreover, the method by which we establish uniqueness of the solution turns out to yield a new completely rigorous derivation of the limit.
Random Shortest Paths: NonEuclidean Instances for Metric Optimization Problems
"... Abstract. Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtai ..."
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Abstract. Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The length of an edge is then the length of a shortest path (with respect to the weights drawn) that connects its two endpoints. We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the kcenter problem, as well as the runningtime of the 2opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worstcase bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons. 1