6SDPBansal - Constructive Algorithms for Discrepancy...

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[cs.DS] 9 Aug 2010 Constructive Algorithms for Discrepancy Minimization Nikhil Bansal * Abstract Given a set system ( V, S ) , V = { 1 ,...,n } and S = { S 1 ,...,S m } , the minimum discrepancy problem is to find a 2-coloring X : V → {− 1 , +1 } , such that each set is colored as evenly as possible, i.e. find X to minimize max j [ m ] v v v i S j X ( i ) v v v . In this paper we give the first polynomial time algorithms for discrepancy minimization that achieve bounds similar to those known existentially using the so-called Entropy Method. We also give a first approximation-like result for discrepancy. Specifically we give efficient randomized algorithms to: 1. Construct an O ( n 1 / 2 ) discrepancy coloring for general sets systems when m = O ( n ) , match- ing the celebrated result of Spencer [17] up to constant factors. Previously, no algorithmic guarantee better than the random coloring bound, i.e. O (( n log n ) 1 / 2 ) , was known. More generally, for m n , we obtain a discrepancy bound of O ( n 1 / 2 log(2 m/n )) . 2. Construct a coloring with discrepancy O ( t 1 / 2 log n ) , if each element lies in at most t sets. This matches the (non-constructive) result of Srinivasan [19]. 3. Construct a coloring with discrepancy O ( λ log( nm )) , where λ is the hereditary discrepancy of the set system. The main idea in our algorithms is to produce a coloring over time by letting the color of the elements perform a random walk (with tiny increments) starting from 0 until they reach 1 or +1 . At each time step the random hops for various elements are correlated using the solution to a semidefinite program, where this program is determined by the current state and the entropy method. 1 Introduction Let ( V, S ) be a set-system, where V = { 1 ,... ,n } are the elements and S = { S 1 ,... ,S m } is a collec- tion of subsets of V . Given a {− 1 , +1 } coloring X of elements in V , let X ( S j ) = i S j X ( i ) denote the discrepancy of X for set S . The discrepancy of the collection S is defined as disc ( S ) = min X max j [ m ] |X ( S j ) | . Understanding the discrepancy of various set-systems has been a major area of research both in math- ematics and computer science, and this study has revealed fascinating connections to various areas of mathematics. Discrepancy also has a range of applications to several topics in computer science such as probabilistic and approximation algorithms, computational geometry, numerical integration, derandom- ization, communication complexity, machine learning, optimization and so on. We shall not attempt to describe these connections and applications here, but refer the reader to [6, 9, 12]. IBM T. J. Watson Research Center, Yorktown Heights, NY 10598. E-mail:
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This note was uploaded on 10/23/2011 for the course CS 7520 taught by Professor Staff during the Spring '08 term at Georgia Tech.

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6SDPBansal - Constructive Algorithms for Discrepancy...

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