[Paper Review] Nearly Linear-Work Algorithms for Mixed Packing/Covering and Facility-Location Linear Programs

Neal E. Young|arXiv (Cornell University)|Jul 11, 2014

Optimization and Search Problems24 references3 citations

TL;DR

This paper presents the first nearly linear-time and nearly linear-work parallel algorithms for mixed packing/covering linear programs and fractional facility location, achieving $(1+\epsilon)$-approximate solutions in $\widetilde{O}(N/\epsilon^2)$ time and work, where $N$ is the number of non-zeros in the constraint matrix. The approach uses a novel Lagrangian relaxation framework with adaptive, non-uniform increments guided by a smooth penalty function.

ABSTRACT

We describe the first nearly linear-time approximation algorithms for explicitly given mixed packing/covering linear programs, and for (non-metric) fractional facility location. We also describe the first parallel algorithms requiring only near-linear total work and finishing in polylog time. The algorithms compute $(1+ε)$-approximate solutions in time (and work) $O^*(N/ε^2)$, where $N$ is the number of non-zeros in the constraint matrix. For facility location, $N$ is the number of eligible client/facility pairs.

Motivation & Objective

• To develop the first nearly linear-time approximation algorithms for explicitly given mixed packing/covering linear programs.
• To design the first parallel algorithms for these problems that achieve near-linear total work and polylogarithmic time.
• To extend nearly linear work and time efficiency to the non-metric fractional facility location problem.
• To overcome the bottleneck of solving LPs in approximation algorithms for NP-hard problems by achieving near-optimal dependence on $\epsilon$ and input size $N$.
• To enable scalable, highly parallelizable solutions for large-scale optimization in big data and multiprocessor environments.

Proposed method

• The algorithm uses a Lagrangian relaxation framework that maintains a smooth penalty function $\phi(x)$ combining packing and covering constraints.
• It incrementally improves the solution by updating variables in directions guided by the gradient of $\phi(x)$, ensuring convergence within a trust region.
• The penalty function is designed as $\phi(x) \approx \log\left(\sum_i (1+\epsilon)^{P_i x} \cdot \sum_i (1-\epsilon)^{C_i x}\right)$ for mixed packing/covering problems.
• For facility location, the algorithm reformulates the problem as a covering LP and applies a specialized variant of the Lagrangian relaxation with adaptive scaling.
• The method uses non-uniform, adaptive increments that depend on the current solution and constraint violations, improving convergence speed.
• The algorithm is parallelized by distributing updates across processors, with work and time bounds derived via careful analysis of iteration counts and data dependencies.

Experimental results

Research questions

• RQ1 Can nearly linear-time algorithms be designed for mixed packing/covering linear programs with $(1+\epsilon)$-approximation?
• RQ2 Can such algorithms be made highly parallel with near-linear total work and polylogarithmic time?
• RQ3 Can the same approach be extended to the non-metric fractional facility location problem?
• RQ4 Can the dependence on $\epsilon$ be kept at $1/\epsilon^2$ while achieving near-linear work and time?
• RQ5 Can the method avoid super-linear blowups in problem size that occur in standard reformulations?

Key findings

• The paper achieves a $(1+\epsilon)$-approximate solution for mixed packing/covering LPs in $O(N \log m / \epsilon^2)$ time, where $N$ is the number of non-zeros in the constraint matrix.
• The parallel algorithm for mixed packing/covering performs $O(N \log m \cdot \log(n \log m / \epsilon) / \epsilon^2)$ work and runs in $O(\log N \cdot \log^2 m \cdot \log(n \log m / \epsilon) / \epsilon^4)$ time.
• For fractional facility location, the sequential algorithm runs in $O(N \log m / \epsilon^2)$ time and the parallel version does $O(N \log N \cdot \log m / \epsilon^2)$ work in polylogarithmic time.
• The algorithms avoid super-linear blowups in problem size by not relying on standard set-cover reformulations, which otherwise increase size non-linearly.
• The method ensures numerical stability and sparsity maintenance, similar to iterative solvers, due to the Lagrangian relaxation framework.
• The algorithm achieves near-optimal $1/\epsilon^2$ dependence on $\epsilon$, improving upon prior work with super-linear dependencies on other parameters.