线性离散的计算复杂性
On the Computational Complexity of Linear Discrepancy
摘要：
计算机科学和应用数学中的许多问题需要将位于区间$[0,1]$中的分数值的向量$\mathbf{w}$舍入为二进制向量$\mathbf{x}$，使得对于给定的矩阵$\mathbf{A}$，$\mathbf{A}\mathbf{x}$尽可能接近$\mathbf{A}\mathbf{w}$。例如，这个问题出现在LP舍入算法用于近似$\mathsf{NP}$ -硬优化问题，并在数值积分的均匀分布的点集的设计。对于一个给定的矩阵$\mathbf{A}$，由最佳可能舍入引起的所有$\mathbf{w}$选择的最坏情况误差由$\mathbf{A}$的线性差异来衡量，这是差异理论中研究的一个量，由Lovasz、Spencer和Vesztergombi（EJC，1986）引入。我们首先研究了线性偏差的计算复杂性。我们的研究在两个方向进行：（1）证明硬度结果和（2）找到精确和近似算法来评估某些矩阵的线性差异。对于（1），我们证明了线性偏差是$\mathsf{NP}$ -困难的。因此，我们不期望找到一个有效的精确算法的一般情况下。限制我们的注意力与一个恒定的行数的矩阵，我们提出了一个多时间精确算法的矩阵组成的一个单一的行和矩阵与一个恒定的行数和有界幅度的条目。我们还提出了一般矩阵的指数时间近似算法，和一个算法，近似的线性偏差内的指数因子。
英文摘要：
Many problems in computer science and applied mathematics require rounding a vector $\mathbf{w}$ of fractional values lying in the interval $[0,1]$ to a binary vector $\mathbf{x}$ so that, for a given matrix $\mathbf{A}$, $\mathbf{A}\mathbf{x}$ is as close to $\mathbf{A}\mathbf{w}$ as possible. For example, this problem arises in LP rounding algorithms used to approximate $\mathsf{NP}$-hard optimization problems and in the design of uniformly distributed point sets for numerical integration. For a given matrix $\mathbf{A}$, the worst-case error over all choices of $\mathbf{w}$ incurred by the best possible rounding is measured by the linear discrepancy of $\mathbf{A}$, a quantity studied in discrepancy theory, and introduced by Lovasz, Spencer, and Vesztergombi (EJC, 1986). We initiate the study of the computational complexity of linear discrepancy. Our investigation proceeds in two directions: (1) proving hardness results and (2) finding both exact and approximate algorithms to evaluate the linear discrepancy of certain matrices. For (1), we show that linear discrepancy is $\mathsf{NP}$-hard. Thus we do not expect to find an efficient exact algorithm for the general case. Restricting our attention to matrices with a constant number of rows, we present a poly-time exact algorithm for matrices consisting of a single row and matrices with a constant number of rows and entries of bounded magnitude. We also present an exponential-time approximation algorithm for general matrices, and an algorithm that approximates linear discrepancy to within an exponential factor.
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