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55
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 sta ..."
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Cited by 56 (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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Frequency Assignment Problems
 HANDBOOK OF COMBINATORIAL OPTIMIZATION
, 1999
"... The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Furt ..."
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Cited by 42 (3 self)
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The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Further at issue are online algorithms for dynamically assigning frequencies to users within an established network. Applications include aeronautical mobile, land mobile, maritime mobile, broadcast, land fixed (pointto point), and satellite systems. This paper surveys research conducted by theoreticians, engineers, and computer scientists regarding the frequency assignment problem (FAP) in all of its guises. The paper begins by defining some of the more common types of FAPs. It continues with a discussion on measures of optimality relating to the use of spectrum, models of interference, and mathematical representations of the many FAPs, both in graph theoretic terms, and as mathematical pro...
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
, 2003
"... 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 prob ..."
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Cited by 20 (1 self)
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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.
The Probabilistic Relationship between the Assignment and Asymmetric Traveling Salesman Problems
, 2001
"... this paper, c0, cl,... are positive absolute constants whose precise values are not too important to us ..."
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Cited by 16 (2 self)
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this paper, c0, cl,... are positive absolute constants whose precise values are not too important to us
Proofs of the Parisi and CoppersmithSorkin random assignment conjectures
, 2005
"... Suppose that there are n jobs and n machines and it costs cij to execute job i on machine j. 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. When the cij are independent and identically distribu ..."
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Cited by 14 (0 self)
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Suppose that there are n jobs and n machines and it costs cij to execute job i on machine j. 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. When the cij are independent and identically distributed exponentials of mean 1, Parisi [Technical Report condmat/9801176, xxx LANL Archive, 1998] made the beautiful conjecture that the expected cost of the minimum assignment equals ∑n i=1 (1/i2). Coppersmith and Sorkin [Random Structures Algorithms 15 (1999), 113–144] generalized Parisi’s conjecture to the average value of the smallest kassignment when there are n jobs and m machines. Building on the previous work of Sharma and Prabhakar [Proc 40th Annu
On the Expected Incremental Cost of a Minimum Assignment
, 1999
"... The random assignment problem is to choose a minimumcost matching in a complete bipartite graph whose edge weights are chosen randomly from some distribution, such as the exponential distribution with parameter 1. When choosing a perfect matching in the complete n \Theta n bipartite graph, it has b ..."
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Cited by 13 (1 self)
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The random assignment problem is to choose a minimumcost matching in a complete bipartite graph whose edge weights are chosen randomly from some distribution, such as the exponential distribution with parameter 1. When choosing a perfect matching in the complete n \Theta n bipartite graph, it has been conjectured that the expected cost is P n i=1 1=i 2 , tending to 2 =6 in the limit. A subsequent, stronger conjecture is that the expectation of a minimumcost matching of cardinality k in a complete m \Theta n bipartite graph is F (k; m;n) j P i;j0; i+j!k 1 (m\Gammai)(n\Gammaj) . In this note we show that, under certain hypotheses, the cost of augmenting a minimum (m \Gamma 1)assignment in an (m \Gamma 1) \Theta n bipartite graph, to a minimum m assignment in an m \Theta n bipartite graph, is equal to F (m; m;n) \Gamma F (m \Gamma 1; m \Gamma 1; n). However, the hypotheses required are such that this result, while intriguing, does not provide a proof of the above conjectur...
A Proof of a Conjecture of Buck, Chan and Robbins on the Expected Value of the Minimum Assignment, Random Structures and Algorithms
 RANDOM STRUCTURES AND ALGORITHMS
, 2005
"... We prove the main conjecture of the paper “On the expected value of the minimum ..."
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Cited by 12 (9 self)
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We prove the main conjecture of the paper “On the expected value of the minimum