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We consider a problem which has received considerable attention in systemsliterature because of its applications to routing in delay tolerant networksand replica placement in distributed storage systems. In abstract terms theproblem can be stated as follows: Given a random variable $X$ generated by aknown product distribution over $\{0,1\}^n$ and a target value $0 \leq \theta\leq 1$, output a non-negative vector $w$, with $\|w\|_1 \le 1$, whichmaximizes the probability of the event $w \cdot X \ge \theta$. This is achallenging 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-boundedproduct distributions, runs in $ \poly(n) \cdot 2^{\poly(1/\eps)}$ time andoutputs an $\eps$-approximately optimal solution vector $w$ for this problem.Our approach is inspired by, and extends, recent structural results from thecomplexity-theoretic study of linear threshold functions. Furthermore, in spiteof the objective function being non-smooth, we give a \emph{unicriterion} PTASwhile previous work for such objective functions has typically led to a\emph{bicriterion} PTAS. We believe our techniques may be applicable to getunicriterion PTAS for other non-smooth objective functions.

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