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A survey of maxtype recursive distributional equations
 Annals of Applied Probability 15 (2005
, 2005
"... In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent cop ..."
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In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write
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 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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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.
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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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 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
Belief propagation: an asymptotically optimal algorithm for the random assignment problem
"... The random assignment problem concerns finding the minimum cost assignment or matching in a complete bipartite graph with edge weights being i.i.d. with some distribution, say exponential(1) distribution. In a remarkable result by Aldous (2001), it was shown that the average cost of such an assignme ..."
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The random assignment problem concerns finding the minimum cost assignment or matching in a complete bipartite graph with edge weights being i.i.d. with some distribution, say exponential(1) distribution. In a remarkable result by Aldous (2001), it was shown that the average cost of such an assignment converges to ζ(2) = π 2 /6 as the size of bipartite graph increases to ∞; thus proving conjecture of Mézard and Parisi (1987) based on replica method arising from statistical physics insights. This conjecture also suggested a heuristic for finding such an assignment, which is an instance of the wellknown heuristic Belief Propagation (BP) discussed by Pearl (1987). In a recent work by Bayati, Shah and Sharma (2005), BP was shown to find correct solution in O(n 3) time for the instance of assignment problem over graph of size n with arbitrary weights. In contrast, in this paper we establish that the BP finds an asymptotically correct assignment in O(n 2) time with high probability for the random assignment problem for a large class of edge weight distributions. Thus, BP is essentially an optimal algorithm for the assignment problem under random setup. Our result utilizes result of Aldous (2001) and the notion of local weak convergence. Key nontrivial steps in establishing our result involve proving attractiveness (aka decay of correlation) of an operator acting on space of distributions corresponding to the mincost matching on Poisson Weighted Infinite Tree (PWIT) and establishing uniform convergence of dynamics of BP on bipartite graph to an appropriately defined dynamics on PWIT. Key words: Belief propagation; random assignment problem; local weak convergence; correlation decay; Poisson weighted infinite tree.
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
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