基于乘性权值更新的确定性离散极小化方法
Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method
摘要：
Spencer的一个著名定理表明，任何在$n$元素上具有$n$集合的集合系统允许差异着色$O(\sqrt{n})$。虽然最初的证明是非建设性的，但最近的进展带来了Bansal，Lovett和Meka以及Rothvoss的多项式时间算法。所有这些算法都是随机的，尽管Bansal的算法承认一个复杂的去随机化。我们提出了一个优雅的确定性的多项式时间算法的灵感来自Lovett-Meka以及乘法权重更新方法。该算法迭代地更新分数着色，同时控制分配给集合约束的指数权重。Meka的一个猜想表明Spencer界可以推广到对称矩阵。我们证明了$n \times n$矩阵是块对角的块大小$q$允许染色的差异$O(\sqrt{n} \cdot \sqrt{\log(q)})$。Bansal，Dadush和Garg最近给出了一个随机算法来在多项式时间内找到一个在$\lbrace{-1,1\rbrace}$中具有$\|Ax\|_{\infty} \leq O(\sqrt{\log n})$的条目的向量$x$，其中$A$是列长度至多为1的任何矩阵。我们表明，我们的方法可以用来确定性地获得这样一个向量。
英文摘要：
A well-known theorem of Spencer shows that any set system with $n$ sets over $n$ elements admits a coloring of discrepancy $O(\sqrt{n})$. While the original proof was non-constructive, recent progress brought polynomial time algorithms by Bansal, Lovett and Meka, and Rothvoss. All those algorithms are randomized, even though Bansal's algorithm admitted a complicated derandomization. We propose an elegant deterministic polynomial time algorithm that is inspired by Lovett-Meka as well as the Multiplicative Weight Update method. The algorithm iteratively updates a fractional coloring while controlling the exponential weights that are assigned to the set constraints. A conjecture by Meka suggests that Spencer's bound can be generalized to symmetric matrices. We prove that $n \times n$ matrices that are block diagonal with block size $q$ admit a coloring of discrepancy $O(\sqrt{n} \cdot \sqrt{\log(q)})$. Bansal, Dadush and Garg recently gave a randomized algorithm to find a vector $x$ with entries in $\lbrace{-1,1\rbrace}$ with $\|Ax\|_{\infty} \leq O(\sqrt{\log n})$ in polynomial time, where $A$ is any matrix whose columns have length at most 1. We show that our method can be used to deterministically obtain such a vector.
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