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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...
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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Cited by 4 (2 self)
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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.
LOCAL TAIL BOUNDS FOR FUNCTIONS OF INDEPENDENT RANDOM VARIABLES
, 2008
"... It is shown that functions defined on {0,1,...,r − 1} n satisfying certain conditions of bounded differences that guarantee subGaussian tail behavior also satisfy a much stronger “local” subGaussian property. For selfbounding and configuration functions we derive analogous locally subexponential ..."
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It is shown that functions defined on {0,1,...,r − 1} n satisfying certain conditions of bounded differences that guarantee subGaussian tail behavior also satisfy a much stronger “local” subGaussian property. For selfbounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand’s [Ann. Probab. 22 (1994) 1576–1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on {0,1,...,r − 1} n for r ≥ 2.
Notes on random optimization problems
, 2008
"... These notes are under construction. They constitute a combination of what I have said in the lectures, what I will say in future lectures, and what I will not say due to time constraints. Some sections are very brief, and this is generally because they are not yet written. Some of the “problems and ..."
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These notes are under construction. They constitute a combination of what I have said in the lectures, what I will say in future lectures, and what I will not say due to time constraints. Some sections are very brief, and this is generally because they are not yet written. Some of the “problems and exercises” describe things that I am actually going to write down in detail in the text. This is because I have used the problems & exercises section in this way to take short notes of things I should not forget to mention.