On the Computational Complexity of Linear Discrepancy

TL;DR

This paper studies the computational complexity of linear discrepancy, proving NP-hardness and developing algorithms for exact and approximate evaluation, with implications for optimization and numerical integration.

Contribution

It establishes the NP-hardness of computing linear discrepancy and provides exact and approximation algorithms for special cases and general matrices.

Findings

Linear discrepancy computation is NP-hard.

Polynomial-time exact algorithm for single-row matrices.

Exponential-time approximation algorithm for general matrices.

Abstract

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.