Sequential Testing with Subadditive Costs

TL;DR

This paper develops approximation algorithms for sequential testing problems with complex subadditive costs, extending classic methods to more general cost structures and providing bounds for various applications.

Contribution

It introduces a novel approximation algorithm for sequential testing with subadditive costs and connects it to the minimum sum set cover problem, covering multiple cost models.

Findings

Achieves a $(4 ho+\gamma)$-approximation under certain oracle assumptions.

Provides the first approximation algorithms for tree-based, routing, and machine activation costs.

Shows hardness results for submodular costs assuming the exponential time hypothesis.

Abstract

In the classic sequential testing problem, we are given a system with several components each of which fails with some independent probability. The goal is to identify whether or not some component has failed. When the test costs are additive, it is well known that a greedy algorithm finds an optimal solution. We consider a much more general setting with subadditive cost functions and provide a $(4\rho+\gamma)$-approximation algorithm, assuming a $\gamma$-approximate value oracle (that computes the cost of any subset) and a $\rho$-approximate ratio oracle (that finds a subset with minimum ratio of cost to failure probability). While the natural greedy algorithm has a poor approximation ratio in the subadditive case, we show that a suitable truncation achieves the above guarantee. Our analysis is based on a connection to the minimum sum set cover problem. As applications, we obtain the first approximation algorithms for sequential testing under various cost-structures: $(5+\epsilon)$-approximation for tree-based costs, $9.5$-approximation for routing costs and $(4+\ln n)$ for machine activation costs. We also show that sequential testing under submodular costs does not admit any poly-logarithmic approximation (assuming the exponential time hypothesis).