Efficient algorithms for discrepancy minimization in convex sets
Efficient algorithms for discrepancy minimization in convex sets
复制标题
凸集差异最小化的有效算法
DOI：
10.1002/rsa.20763
复制
发表时间：
2014
影响因子：
1
通讯作者：
Mohit Singh
中科院分区：
文献类型：
--
作者：
Ronen Eldan;Mohit Singh
Spencer的结果指出，N大小N的每个集合的n集合具有{-1，+1}差异O（N）的地面集的着色。另请参见Giannopoulos），他表明每个对称凸体k⊆rn具有高斯测量至少E -ϵn的每个对称凸体，对于一个小的ϵ> 0，包含一个点y∈K，其中y的恒定分数y在{-1中均在{-1中， 1} 。与Gluskin和Giannopoulos的结果相同。和有效的算法，我们首先证明了Gluskin和Giannopoulos结果的另一个建设性版本，其中着色是通过线性函数的优化来实现的。作为Spencer。 （可能是非对称的）凸体k⊆rn，具有高斯测量至少E -ϵn，对于一个小的ϵ> 0，包含一个点y∈K，其中y的坐标的恒定分数在{-1,1}中最后，我们给出了一个简单的证据，表明对任何δ> 0都存在一个常数C> 0在CK中，这给出了Banaszczyk结果的特殊情况的算法版本。
A result of Spencer states that every collection of n sets over a universe of size n has a coloring of the ground set with {−1,+1} of discrepancy O(n) . A geometric generalization of this result was given by Gluskin (see also Giannopoulos) who showed that every symmetric convex body K⊆Rn with Gaussian measure at least e−ϵn , for a small ϵ>0 , contains a point y∈K where a constant fraction of coordinates of y are in {−1,1} . This is often called a partial coloring result. While the proofs of both these results were inherently non‐algorithmic, recently Bansal (see also Lovett‐Meka) gave a polynomial time algorithm for Spencer's setting and Rothvoß gave a randomized polynomial time algorithm obtaining the same guarantee as the result of Gluskin and Giannopoulos. This paper contains several related results which combine techniques from convex geometry to analyze simple and efficient algorithms for discrepancy minimization. First, we prove another constructive version of the result of Gluskin and Giannopoulos, in which the coloring is attained via the optimization of a linear function. This implies a linear programming based algorithm for combinatorial discrepancy obtaining the same result as Spencer. Our second result suggests a new approach to obtain partial colorings, which is also valid for the non‐symmetric case. It shows that every (possibly non‐symmetric) convex body K⊆Rn , with Gaussian measure at least e−ϵn , for a small ϵ>0 , contains a point y∈K where a constant fraction of coordinates of y are in {−1,1} . Finally, we give a simple proof that shows that for any δ>0 there exists a constant c > 0 such that given a body K with γn(K)≥δ , a uniformly random x from {−1,1}n is in cK with constant probability. This gives an algorithmic version of a special case of the result of Banaszczyk.