A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs

TL;DR

This paper presents a nearly linear-time approximation scheme for solving packing and covering linear programs with guarantees on solution quality, significantly improving computational efficiency for large sparse problems.

Contribution

It introduces a nearly linear-time PTAS for explicit fractional packing and covering linear programs, optimizing solution computation for large sparse matrices.

Findings

Achieves a (1+eps)-approximate solution in near-linear time

Efficiently handles large sparse matrices with non-zero entries

Provides feasible primal and dual solutions close to optimal

Abstract

We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm computes feasible primal and dual solutions whose costs are within a factor of 1+eps of the optimal cost in time O((r+c)log(n)/eps^2 + n).