Max-Throughput for (Conservative) k-of-n Testing

We define a variant of k-of-n testing that we call conservative k-of-n testing. We present a polynomial-time, combinatorial algorithm for the problem of maximizing throughput of conservative k-of-n testing, in a parallel setting. This extends previous work of Kodialam and Condon et al., who presented combinatorial algorithms for parallel pipelined filter ordering, which is the special case where k=1 (or k = n). We also consider the problem of maximizing throughput for standard k-of-n testing, and show how to obtain a polynomial-time algorithm based on the ellipsoid method using previous techniques.

💡 Research Summary

The paper introduces a new variant of the classic k‑of‑n testing problem, called conservative k‑of‑n testing, and studies its maximum‑throughput (Max‑Throughput) optimization in a parallel processing environment. In standard k‑of‑n testing an item is examined sequentially until either k failures (0’s) are observed or n‑k+1 successes (1’s) are observed, at which point the decision can be made and testing stops. In the conservative variant the process continues until either k failures are observed or all n tests have been performed, regardless of how many successes have already appeared. This distinction matters in settings where, for items that do not fail enough tests to be discarded, the exact set of failed tests must still be identified (e.g., quality‑control repair, medical diagnostics that need to locate the specific abnormal tests).

The authors model a system where each test i is executed by a dedicated processor O_i with a rate limit r_i (items per unit time). Items arrive as an infinite stream; each item is assigned a testing strategy T (a permutation of the tests together with a stopping rule). The strategy determines which tests are actually performed on that item, and the probability that test i is performed under strategy T is denoted g(T,i). A routing assigns a flow value z_T to each strategy, representing how many items per unit time follow that strategy. The linear program (LP) for Max‑Throughput is:

max   F = Σ_T z_T
s.t.  Σ_T g(T,i)·z_T ≤ r_i   for all i
      z_T ≥ 0

The goal is to maximize the total throughput F while respecting the per‑processor capacity constraints.

Main Theoretical Insight for Conservative k‑of‑n

A key lemma (Lemma 1) shows that if a routing saturates every processor (i.e., each constraint is tight), then the routing is optimal. The proof hinges on the fact that, in the conservative model, the expected number of “0‑valued” test outcomes generated per unit time is fixed for any routing with a given throughput F. Specifically, each processor i can generate at most r_i·(1‑p_i) zero outcomes per unit time, where p_i is the probability that test i returns 1. The total number of zero outcomes generated by the whole system must equal F·M, where

M = k·Pr