Efficient Redundant Assignments under Fault-Tolerance Constraints

Abstract

We consider the problem of computing minimum congestion, fault-tolerant, redundant assignments of messages to faulty parallel delivery channels. In particular, we are given a set M of faulty channels, each having an integer capacity c _i and failing independently with probability f _i . We are also given a set of messages to be delivered over M, and a fault-tolerance constraint (1– ε), and we seek a redundant assignment φ that minimize congestion Cong(φ), i.e. the maximum channel load, subject to the constraint that, with probability no less than (1– ε), all the messages have a copy on at least one active channel. We present a 4-approximation algorithm for identical capacity channels and arbitrary messages sizes, and a $2 \left \lceil \frac{\ln(\vert M\vert/\epsilon)}{\ln(1/f_{\mathrm{max}})} \right \rceil$-approximation algorithm for related capacity channels and unit size messages.

Both algorithms are based on computing a collection of disjoint channel subsets such that, with probability no less than (1– ε), at least one channel is active in each subset. The objective is to maximize the sum of the minimum subset capacities. Since the exact version of this problem is $\mathcal{NP}$-complete, we present a 2-approximation algorithm for identical capacities, and a (8 + o(1))-approximation algorithm for arbitrary capacities.

This work was partially supported by ESPRIT LTR Project no. 20244—ALCOM-IT.