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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...
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
A simple proof of the Parisi and COPPERSMITHSORKIN FORMULAS FOR THE RANDOM ASSIGNMENT PROBLEM
 LINKÖPING STUDIES IN MATHEMATICS, NO. 6
, 2005
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Exact formulas and limits for a class of random optimization problems
 LINKÖPING STUDIES IN MATHEMATICS
, 2005
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Edge cover and polymatroid flow problems
, 2010
"... In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This socalled minimum edge cover problem is a relaxation of perfect matching. We show that the large n l ..."
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Cited by 4 (1 self)
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In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This socalled minimum edge cover problem is a relaxation of perfect matching. We show that the large n limit cost of the minimum edge cover is W (1) 2 +2W (1) ≈ 1.456, where W is the Lambert Wfunction. In particular this means that the minimum edge cover is essentially cheaper than the minimum perfect matching, whose limit cost is π 2 /6 ≈ 1.645. We obtain this result through a generalization of the perfect matching problem to a setting where we impose a (poly)matroid structure on the two vertexsets of the graph, and ask for an edge set of prescribed size connecting independent sets.
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
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.