CONSTRUCTIVE DISCREPANCY MINIMIZATION BY WALKING ON THE EDGES

被引:43
|
作者
Lovett, Shachar [1 ]
Meka, Raghu [2 ]
机构
[1] Univ Calif San Diego, La Jolla, CA 92093 USA
[2] Microsoft Res, Mountain View, CA 94043 USA
关键词
discrepancy; random walks; Brownian motion;
D O I
10.1137/130929400
中图分类号
TP301 [理论、方法];
学科分类号
081202 ;
摘要
Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer [Trans. Amer. Math. Soc., 289 (1985), pp. 679-706]: In any system of n sets in a universe of size n, there always exists a coloring which achieves discrepancy 6 root n. The original proof of Spencer was existential in nature and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal [Proceedings of FOCS, 2010, pp. 3-10] gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is truly constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.
引用
收藏
页码:1573 / 1582
页数:10
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