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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 41 (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
Efficient algorithms for threedimensional axial and planar random assignment problems
, 2013
"... Beautiful formulas are known for the expected cost of random twodimensional assignment problems, but in higher dimensions, even the scaling is not known. In three dimensions and above, the problem has natural “Axial ” and “Planar ” versions, both of which are NPhard. For 3dimensional Axial random ..."
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Cited by 3 (1 self)
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Beautiful formulas are known for the expected cost of random twodimensional assignment problems, but in higher dimensions, even the scaling is not known. In three dimensions and above, the problem has natural “Axial ” and “Planar ” versions, both of which are NPhard. For 3dimensional Axial random assignment instances of size n, the cost scales as Ω(1/n), and a main result of the present paper is a lineartime algorithm that, with high probability, finds a solution of cost O(n −1+o(1)). For 3dimensional Planar assignment, the lower bound is Ω(n), and we give a new efficient matchingbased algorithm that with high probability returns a solution with cost O(n log n). 1
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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Cited by 2 (0 self)
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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.
Minimumcost matching in a random graph with random costs
, 2015
"... Let Gn,p be the standard ErdősRényiGilbert random graph and let Gn,n,p be the random bipartite graph on n + n vertices, where each e ∈ [n]2 appears as an edge independently with probability p. For a graph G = (V,E), suppose that each edge e ∈ E is given an independent uniform exponential rate on ..."
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Let Gn,p be the standard ErdősRényiGilbert random graph and let Gn,n,p be the random bipartite graph on n + n vertices, where each e ∈ [n]2 appears as an edge independently with probability p. For a graph G = (V,E), suppose that each edge e ∈ E is given an independent uniform exponential rate one cost. Let C(G) denote the random variable equal to the length of the minimum cost perfect matching, assuming that G contains at least one. We show that w.h.p. if d = np (log n)2 then w.h.p. E [C(Gn,n,p)] = (1 + o(1))pi26p. This generalises the wellknown result for the case G = Kn,n. We also show that w.h.p. E [C(Gn,p)] = (1 + o(1)) pi2 12p along with concentration results for both types of random graph. 1
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.