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Partitioning Problem (partitioning + problem)

Distribution by Scientific Domains

Mathematics and Statistics	75%

Selected Abstracts

Approximation algorithms for channel allocation problems in broadcast networks

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2006
Rajiv Gandhi
Abstract We study two packing problems that arise in the area of dissemination-based information systems; a second theme is the study of distributed approximation algorithms. The problems considered have the property that the space occupied by a collection of objects together could be significantly less than the sum of the sizes of the individual objects. In the Channel Allocation Problem, there are requests that are subsets of topics. There are a fixed number of channels that can carry an arbitrary number of topics. All the topics of each request must be broadcast on some channel. The load on any channel is the number of topics that are broadcast on that channel; the objective is to minimize the maximum load on any channel. We present approximation algorithms for this problem, and also show that the problem is MAX-SNP hard. The second problem is the Edge Partitioning Problem addressed by Goldschmidt, Hochbaum, Levin, and Olinick (Networks, 41:13,23, 2003). Each channel here can deliver topics for at most k requests, and we aim to minimize the total load on all channels. We present an O(n1/3),approximation algorithm, and also show that the algorithm can be made fully distributed with the same approximation guarantee; we also generalize the (nondistributed) Edge Partitioning Problem of graphs to the case of hypergraphs. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(4), 225,236 2006 [source]

Time , size tradeoffs: a phylogenetic comparative study of flowering time, plant height and seed mass in a north-temperate flora

OIKOS, Issue 3 2008
Kjell Bolmgren
Parents face a timing problem as to when they should begin devoting resources from their own growth and survival to mating and offspring development. Seed mass and number, as well as maternal survival via plant size, are dependent on time for development. The time available in the favorable season will also affect the size of the developing juveniles and their survival through the unfavorable season. Flowering time may thus represent the outcome of such a time partitioning problem. We analyzed correlations between flowering onset time, seed mass, and plant height in a north-temperate flora, using both cross-species comparisons and phylogenetic comparative methods. Among perennial herbs, flowering onset time was negatively correlated with seed mass (i.e. plants with larger seeds started flowering earlier) while flowering onset time was positively correlated with plant height. Neither of these correlations was found among woody plants. Among annual plants, flowering onset time was positively correlated with seed mass. Cross-species and phylogenetically informed analyses largely agreed, except that flowering onset time was also positively correlated with plant height among annuals in the cross-species analysis. The different signs of the correlations between flowering onset time and seed mass (compar. gee regression coefficient=,7.8) and flowering onset time and plant height (compar. gee regression coefficient=+30.5) for perennial herbs, indicate that the duration of the growth season may underlie a tradeoff between maternal size and offspring size in perennial herbs, and we discuss how the partitioning of the season between parents and offspring may explain the association between early flowering and larger seed mass among these plants. [source]

Poisson convergence in the restricted k -partitioning problem

RANDOM STRUCTURES AND ALGORITHMS, Issue 4 2007
Anton Bovier
Abstract The randomized k -number partitioning problem is the task to distribute N i.i.d. random variables into k groups in such a way that the sums of the variables in each group are as similar as possible. The restricted k -partitioning problem refers to the case where the number of elements in each group is fixed to N/k. In the case k = 2 it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case k > 2 in the restricted problem and show that the vector of differences between the k sums converges to a k - 1-dimensional Poisson point process. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 [source]

Phase transition and finite-size scaling for the integer partitioning problem

RANDOM STRUCTURES AND ALGORITHMS, Issue 3-4 2001
Christian Borgs
We consider the problem of partitioning n randomly chosen integers between 1 and 2m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of ,=m/n, we prove that the problem has a phase transition at ,=1, in the sense that for ,<1, there are many perfect partitions with probability tending to 1 as n,,, whereas for ,>1, there are no perfect partitions with probability tending to 1. Moreover, we show that this transition is first-order in the sense the derivative of the so-called entropy is discontinuous at ,=1. We also determine the finite-size scaling window about the transition point: ,n=1,(2n),1,log2,n+,n/n, by showing that the probability of a perfect partition tends to 1,,0, or some explicitly computable p(,),(0,,1), depending on whether ,n tends to ,,,,,, or ,,(,,,,,), respectively. For ,n,,, fast enough, we show that the number of perfect partitions is Gaussian in the limit. For ,n,,, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is ,(2). Within the window, i.e., if |,n| is bounded, we prove that the optimum discrepancy is bounded. Both for ,n,, and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the k smallest discrepancies above the scaling window. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 247,288, 2001 [source]

Very large-scale neighborhood search

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, Issue 4-5 2000
R.K. Ahuja
Abstract Neighborhood search algorithms are often the most effective approaches available for solving partitioning problems, a difficult class of combinatorial optimization problems arising in many application domains including vehicle routing, telecommunications network design, parallel machine scheduling, location theory, and clustering. A critical issue in the design of a neighborhood search algorithm is the choice of the neighborhood structure, that is, the manner in which the neighborhood is defined. Currently, the two-exchange neighborhood is the most widely used neighborhood for solving partitioning problems. The paper describes the cyclic exchange neighborhood, which is a generalization of the two-exchange neighborhood in which a neighbor is obtained by performing a cyclic exchange. The cyclic exchange neighborhood has substantially more neighbors compared to the two-exchange neighborhood. This paper outlines a network optimization based methodology to search the neighborhood efficiently and presents a proof of concept by applying it to the capacitated minimum spanning tree problem, an important problem in telecommunications network design. [source]