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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...
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
Random assignment and shortest path problems
, 2006
"... We explore a similarity between the n by n random assignment problem and the random shortest path problem on the complete graph on n + 1 vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We g ..."
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Cited by 10 (2 self)
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We explore a similarity between the n by n random assignment problem and the random shortest path problem on the complete graph on n + 1 vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his ζ(2) limit theorem for the assignment problem.
A simple proof of the Parisi and COPPERSMITHSORKIN FORMULAS FOR THE RANDOM ASSIGNMENT PROBLEM
 LINKÖPING STUDIES IN MATHEMATICS, NO. 6
, 2005
"... ..."
On the Random 2Stage Minimum Spanning Tree
, 2004
"... It is known [6] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to (3) = . Here we consider the following stochastic twostage version of this op ..."
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Cited by 9 (3 self)
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It is known [6] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to (3) = . Here we consider the following stochastic twostage version of this optimization problem. There are two sets of edge costs c M : E ! R and c T : E ! R , called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs c M (e) and c T (e) are independent random variables, uniformly distributed in [0; 1]. The Monday costs are revealed rst. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c M (e), or to wait until its Tuesday price c T (e) appears. The set of edges XM bought on Monday is then completed by the set of edges X T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost (3)=2 + o(1). We show that in the case of twostage optimization, the expected value of the optimal cost exceeds (3)=2 by an absolute constant > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than and completes them on Tuesday in an optimal way, and show that the optimal choice for is = 1=n with the expected cost (3) 1=2 + o(1). The threshold heuristic is shown to be suboptimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning outarborescence rooted at a xed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 1=e + o(1).
Exact formulas and limits for a class of random optimization problems
 LINKÖPING STUDIES IN MATHEMATICS
, 2005
"... ..."
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.