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On Generalizations of Network Design Problems with Degree Bounds
, 2010
"... Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum sp ..."
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Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely laminar crossing spanning tree), and (2) by incorporating ‘degree bounds ’ in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems. • Our main result is a (1, b+O(log n))approximation algorithm for the minimum crossing spanning tree (MCST) problem with laminar degree constraints. The laminar MCST problem is a natural generalization of the wellstudied boundeddegree MST, and is a special case of general crossing spanning tree. We give an additive Ω(log α m) hardness of approximation for general MCST, even in the absence of costs (α> 0 is a fixed constant, and m is the number of degree constraints). This also leads to a multiplicative Ω(log α m) hardness of approximation for the robust kmedian problem [1], improving over the previously known factor 2 hardness. • We then consider the crossing contrapolymatroid intersection problem and obtain a (2, 2b + ∆−1)approximation algorithm, where ∆ is the maximum element frequency. This models for example the degreebounded spanningset intersection in two matroids. Finally, we introduce the crossing lattice polyhedra problem, and obtain a (1, b + 2 ∆ − 1) approximation under certain condition. This result provides a unified framework and common generalization of various problems studied previously, such as degree bounded matroids.
Mergeable Summaries
"... We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two data sets, there is a way to merge the two summaries into a single summary on the union of the two data sets, while preserving the error and size guarantees. This property means t ..."
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Cited by 4 (0 self)
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We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two data sets, there is a way to merge the two summaries into a single summary on the union of the two data sets, while preserving the error and size guarantees. This property means that the summaries can be merged in a way like other algebraic operators such as sum and max, which is especially useful for computing summaries on massive distributed data. Several data summaries are trivially mergeable by construction, most notably all the sketches that are linear functions of the data sets. But some other fundamental ones like those for heavy hitters and quantiles, are not (known to be) mergeable. In this paper, we demonstrate that these summaries are indeed mergeable or can be made mergeable after appropriate modifications. Specifically, we show that for εapproximate heavy hitters, there is a deterministic mergeable summary of size O(1/ε); for εapproximate quantiles, there is a deterministic summary of size O ( 1 log(εn)) that has a restricted form of mergeability, ε and a randomized one of size O ( 1 1 log3/2) with full mergeε ε ability. We also extend our results to geometric summaries such as εapproximations and εkernels. We also achieve two results of independent interest: (1) we provide the best known randomized streaming bound for εapproximate quantiles that depends only on ε, of size O ( 1 1 log3/2), and (2) we demonstrate that the MG and the ε ε SpaceSaving summaries for heavy hitters are isomorphic. Supported by NSF under grants CNS0540347, IIS07
The Dawn of an Algebraic . . .
, 2011
"... To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers. Of course, hardly any of these great results have com ..."
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To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers. Of course, hardly any of these great results have come out of the blue: usually the paper I refer to adds the last step to earlier ideas. Since this is an extended abstract (of a nonexistent paper), I will be rather brief, or sometimes completely silent, about the history, with apologies to the unmentioned giants on whose shoulders the authors I do mention have been standing. 1 A careful reader may notice that together with these great results, I will also advertise some smaller results of mine. • Larry Guth and Nets Hawk Katz [16] completed a bold project of György Elekes (whose previous stage is reported in [10]) and obtained a neartight bound for the Erdős distinct distances problem: they proved that every n points in the plane determine at least Ω(n / log n) distinct distances. This almost matches the best known upper bound of O(n / √ √ √ log n), attained for the n × n grid. Their proof and some related results and methods constitute the main topic of this note, and will be discussed later. • János Pach and Gábor Tardos [27] found tight lower bounds for the size of εnets for geometric set systems. 2 It has been known for a long time
and
"... A well studied special case of bin packing is the 3partition problem, where n items of size> 1 4 have to be packed in a minimum number of bins of capacity one. The famous KarmarkarKarp algorithm transforms a fractional solution of a suitable LP relaxation for this problem into an integral solution ..."
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A well studied special case of bin packing is the 3partition problem, where n items of size> 1 4 have to be packed in a minimum number of bins of capacity one. The famous KarmarkarKarp algorithm transforms a fractional solution of a suitable LP relaxation for this problem into an integral solution that requires at most O(logn) additional bins. The threepermutationsproblem of Beck is the following. Given any 3 permutations on n symbols, color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols is roughly the same. The necessary difference is called the discrepancy. We establish a surprising connection between bin packing and Beck’s problem: The additive integrality gap of the 3partition linear programming relaxation can be bounded by the discrepancy of 3 permutations. This connection yields an alternative method to establish an O(logn) bound on the additive integrality gap of the 3partition. Reversely, making use of a recent example of 3 permutations, for which a discrepancy of Ω(logn) is necessary, we prove the following: The O(log 2 n) upper bound on the additive gap for bin packing with arbitrary item sizes cannot be improved by any technique that isbased on rounding up items. Thislower bound holdsfor a large classof algorithms including the KarmarkarKarp procedure.