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On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts  Towards Memetic Algorithms
, 1989
"... Short abstract, isn't it? P.A.C.S. numbers 05.20, 02.50, 87.10 1 Introduction Large Numbers "...the optimal tour displayed (see Figure 6) is the possible unique tour having one arc fixed from among 10 655 tours that are possible among 318 points and have one arc fixed. Assuming that one could ..."
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Cited by 186 (10 self)
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Short abstract, isn't it? P.A.C.S. numbers 05.20, 02.50, 87.10 1 Introduction Large Numbers "...the optimal tour displayed (see Figure 6) is the possible unique tour having one arc fixed from among 10 655 tours that are possible among 318 points and have one arc fixed. Assuming that one could possibly enumerate 10 9 tours per second on a computer it would thus take roughly 10 639 years of computing to establish the optimality of this tour by exhaustive enumeration." This quote shows the real difficulty of a combinatorial optimization problem. The huge number of configurations is the primary difficulty when dealing with one of these problems. The quote belongs to M.W Padberg and M. Grotschel, Chap. 9., "Polyhedral computations", from the book The Traveling Salesman Problem: A Guided tour of Combinatorial Optimization [124]. It is interesting to compare the number of configurations of realworld problems in combinatorial optimization with those large numbers arising in Cosmol...
The objective method: Probabilistic combinatorial optimization and local weak convergence
, 2003
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A survey of maxtype recursive distributional equations
 Annals of Applied Probability 15 (2005
, 2005
"... In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent cop ..."
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Cited by 63 (6 self)
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In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write
Constructive Bounds and Exact Expectations for the Random Assignment Problem
, 1998
"... The random assignment problem is to choose a minimumcost perfect matching in a complete n x n bipartite graph, whose edge weights are chosen randomly from some distribution such as the exponential distribution with mean 1. In this case it is known that the expectation does not grow unboundedly with ..."
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Cited by 49 (5 self)
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The random assignment problem is to choose a minimumcost perfect matching in a complete n x n bipartite graph, whose edge weights are chosen randomly from some distribution such as the exponential distribution with mean 1. In this case it is known that the expectation does not grow unboundedly with n, but approaches some limiting value c ? between 1.51 and 2. The limit is conjectured to be ß 2 =6, while a recent conjecture has it that for finite n, the expected cost is P n i=1 1=i 2 . This paper contains two principal results. First, by defining and analyzing a constructive algorithm, we show that the limiting expectation is c ? ! 1:94. Second, we extend the finiten conjecture to partial assignments on complete m x n bipartite graphs, and prove it in some limited cases.
The ζ(2) Limit in the Random Assignment Problem
, 2000
"... The random assignment (or bipartite matching) problem asks about An = min P n i=1 c(i; (i)), where (c(i; j)) is a n \Theta n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations . M'ezard and Parisi (1987) used the replica method from statisti ..."
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Cited by 40 (1 self)
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The random assignment (or bipartite matching) problem asks about An = min P n i=1 c(i; (i)), where (c(i; j)) is a n \Theta n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations . M'ezard and Parisi (1987) used the replica method from statistical physics to argue nonrigorously that EAn ! i(2) = 2 =6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the i(2) limit and of the conjectured limit distribution of edgecosts and their rankorders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almostoptimal matching coincides with the optimal matching except on a small proportion of edges. Key words and phrases. Assignment problem, bipartite matching, cavity method, combinatorial optimization, distributional identity, infinite tree, probabilistic a...
A proof of Parisi’s conjecture on the random assignment problem
 PROBAB. THEORY RELAT. FIELDS
, 2003
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Threshold Phenomena in Random Graph Colouring and Satisfiability
, 1999
"... We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k ..."
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Cited by 24 (4 self)
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We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k (n, d) = Pr[G(n, d/n) is kcolourable]. Erdos asked the following fundamental question: for k 3, is there a constant c k such that for any # > 0, #) = 1 , and lim f k (n, c k + #) = 0 ? (1) We prove that for all k 3, there exists a function t k (n) such that (1) holds upon replacing c k by t k (n), thus establishing that indeed kcolourability has a sharp threshold. Let d k = sup{d lim n## f k (n, d) = 1}. Note that if c k exists then, by definition, c k = d k . For the basic and most studied case k = 3 we prove 3.84 < d 3 < 5.05 . These are the best
Exact Expectations and Distributions or the Random Assignment Problem
, 1999
"... A generalization of the random assignment problem asks the expected cost of the minimumcost matching of cardinality k in a complete bipartire graph Kmr*, with independent random edge weights. With weights drawn from the exponential(l) distribution, the answer has been conjectured 1 to be ]1,5_ ..."
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Cited by 20 (0 self)
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A generalization of the random assignment problem asks the expected cost of the minimumcost matching of cardinality k in a complete bipartire graph Kmr*, with independent random edge weights. With weights drawn from the exponential(l) distribution, the answer has been conjectured 1 to be ]1,5_>0, i+5< k (i)('5) ' Here, we prove the conjecture for k < 4, k = rn = 5, and k = rn = n = 6, using a structured, automated proof technique that results in proofs with relatively few cases. The method yields not only the minimum assignment cost's expectation but the Laplace transform of its distribution as well. From the Laplace transform we compute the variance in these cases, and conjecture that with k = rn = n  e<>, the variance is 2/n+ O (log n/n 2 ). We also include some asymptotic properties of the expectation and variance when k is fixed.
Proofs of the Parisi and CoppersmithSorkin conjectures for the finite random assignment problem. http://www.stanford.edu/˜ mchandra
, 2003
"... made the following conjecture: Suppose that there are jobs and machines and it costs to execute job on machine. The assignment problem concerns the determination of a onetoone assignment of jobs onto machines so as to minimize the cost of executing all the jobs. The average case analysis of the cl ..."
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Cited by 18 (1 self)
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made the following conjecture: Suppose that there are jobs and machines and it costs to execute job on machine. The assignment problem concerns the determination of a onetoone assignment of jobs onto machines so as to minimize the cost of executing all the jobs. The average case analysis of the classical random assignment problem has received a lot of interest in the recent literature, mainly due to the following pleasing conjecture of Parisi: The average value of the minimumcost permutation in an matrix with i.i.d. entries equals. Coppersmith and Sorkin (1999) have generalized Parisi’s conjecture to the average value of the smallestassignment when there are jobs and machines. We prove both conjectures based on a common set of combinatorial and probabilistic arguments. 1.