A Polynomial-time Approximation Scheme for Fault-tolerant Distributed Storage

TL;DR

This paper presents a polynomial-time approximation scheme for a complex non-convex optimization problem in distributed storage, achieving near-optimal solutions efficiently for certain distributions.

Contribution

It introduces an additive EPTAS for a challenging probabilistic optimization problem, extending structural results from linear threshold functions and providing a unicriterion PTAS for non-smooth objectives.

Findings

Provides an additive EPTAS with polynomial runtime for constant-bounded distributions.

Achieves a unicriterion PTAS for a non-smooth objective function.

Extends structural results from complexity theory to practical approximation algorithms.

Abstract

We consider a problem which has received considerable attention in systems literature because of its applications to routing in delay tolerant networks and replica placement in distributed storage systems. In abstract terms the problem can be stated as follows: Given a random variable $X$ generated by a known product distribution over $\{0,1\}^n$ and a target value $0 \leq \theta \leq 1$, output a non-negative vector $w$, with $\|w\|_1 \le 1$, which maximizes the probability of the event $w \cdot X \ge \theta$. This is a challenging non-convex optimization problem for which even computing the value $\Pr[w \cdot X \ge \theta]$ of a proposed solution vector $w$ is #P-hard. We provide an additive EPTAS for this problem which, for constant-bounded product distributions, runs in $ \poly(n) \cdot 2^{\poly(1/\eps)}$ time and outputs an $\eps$-approximately optimal solution vector $w$ for this problem. Our approach is inspired by, and extends, recent structural results from the complexity-theoretic study of linear threshold functions. Furthermore, in spite of the objective function being non-smooth, we give a \emph{unicriterion} PTAS while previous work for such objective functions has typically led to a \emph{bicriterion} PTAS. We believe our techniques may be applicable to get unicriterion PTAS for other non-smooth objective functions.