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On kColumn Sparse Packing Programs
, 908
"... We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k ..."
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Cited by 11 (3 self)
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We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k2) [3, 5]. We also show that the integrality gap of our linear programming relaxation is at least 2k − 1; it is known that kcolumn sparse PIPs are Ω ( k log k)hard to approximate [8]. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over kcolumn sparse packing constraints. 1
Approximation of MultiColor Discrepancy
 Randomization, Approximation and Combinatorial Optimization (Proceedings of APPROXRANDOM 1999), volume 1671 of Lecture Notes in Computer Science
, 1999
"... . In this article we introduce (combinatorial) multicolor discrepancy and generalize some classical results from 2color discrepancy theory to c colors. We give a recursive method that constructs ccolorings from approximations to the 2color discrepancy. This method works for a large class of ..."
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Cited by 9 (8 self)
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. In this article we introduce (combinatorial) multicolor discrepancy and generalize some classical results from 2color discrepancy theory to c colors. We give a recursive method that constructs ccolorings from approximations to the 2color discrepancy. This method works for a large class of theorems like the sixstandarddeviation theorem of Spencer, the BeckFiala theorem and the results of Matousek, Welzl and Wernisch for bounded VCdimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of twocolor discrepancy even if c is a power of 2. For the linear discrepancy version of the BeckFiala theorem the recursive approach also fails. Here we extend the method of floating colors to multicolorings and prove multicolor versions of the the BeckFiala theorem and the BaranyGrunberg theorem. 1 Introduction Combinatorial discrepancy theory deals with the problem of partitioning the vertices of a hypergraph (set...
Roundings respecting hard constraints
 In Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science
, 2005
"... Abstract. A problem arising in integer linear programming is to transform a solution of a linear system to an integer one which is “close”. The customary model to investigate such problems is, given a matrix A and a [0, 1] valued vector x, to find a binary vector y such that �A(x − y)�∞ (the violati ..."
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Cited by 5 (3 self)
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Abstract. A problem arising in integer linear programming is to transform a solution of a linear system to an integer one which is “close”. The customary model to investigate such problems is, given a matrix A and a [0, 1] valued vector x, to find a binary vector y such that �A(x − y)�∞ (the violation of the constraints) is small. Randomized rounding and the algorithm of Beck and Fiala are ways to compute such solutions y, whereas linear discrepancy is a lower bound measure. In many applications one is looking for roundings that, in addition to being close to the original solution, satisfy some constraints without violation. The objective of this paper is to investigate such problems in a unified way. To this aim, we extend the notion of linear discrepancy, the theorem of Beck and Fiala and the method of randomized rounding to this setting. Whereas some of our examples show that additional hard constraints may seriously increase the linear discrepancy, the latter two sets of results demonstrate that a reasonably broad notion of hard constraints may be added to the rounding problem without worsening the obtained solution significantly. Of particular interest might be our results on randomized rounding. We provide a simpler way to randomly round fixed weight vectors (cf. Srinivasan, FOCS 2001). It has the additional advantage that it can be derandomized with standard methods.
THE HEREDITARY DISCREPANCY IS NEARLY INDEPENDENT OF THE NUMBER OF COLORS
"... Abstract. We investigate the discrepancy (or balanced coloring) problem for hypergraphs and matrices in arbitrary numbers of colors. We show that the hereditary discrepancy in two different numbers a, b ∈ N≥2 of colors is the same apart from constant factors, i.e., herdisc(·,b)=Θ(herdisc(·,a)). This ..."
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Cited by 1 (1 self)
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Abstract. We investigate the discrepancy (or balanced coloring) problem for hypergraphs and matrices in arbitrary numbers of colors. We show that the hereditary discrepancy in two different numbers a, b ∈ N≥2 of colors is the same apart from constant factors, i.e., herdisc(·,b)=Θ(herdisc(·,a)). This contrasts the ordinary discrepancy problem, where no correlation exists in many cases. 1. Introduction and
FLAT POLYNOMIALS ON THE UNIT CIRCLE—NOTE ON A PROBLEM OF LITTLEWOOD
"... As in Littlewood [10], we let #" „ be the class of polynomials of the form where ek = ± 1, and ( Sn be the class of fc0 with coefficients of modulus one, that is ak = e 2nitXk, 0 ^ a ^ < 1 real constants. On the ..."
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Cited by 1 (0 self)
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As in Littlewood [10], we let #" „ be the class of polynomials of the form where ek = ± 1, and ( Sn be the class of fc0 with coefficients of modulus one, that is ak = e 2nitXk, 0 ^ a ^ < 1 real constants. On the
Hereditary Discrepancies in Different Numbers of Colors II
, 2008
"... We bound the hereditary discrepancy of a hypergraph H in two colors in terms of its hereditary discrepancy in c colors. We show that herdisc(H,2) ≤ Kcherdisc(H,c), where K is some absolute constant. This bound is sharp. 1 ..."
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We bound the hereditary discrepancy of a hypergraph H in two colors in terms of its hereditary discrepancy in c colors. We show that herdisc(H,2) ≤ Kcherdisc(H,c), where K is some absolute constant. This bound is sharp. 1