TY - JOUR

T1 - Generalized framework for Group Testing

T2 - Queries, feedbacks and adversaries

AU - Klonowski, Marek

AU - Kowalski, Dariusz R.

AU - Pająk, Dominik

N1 - Funding Information:
Partially supported by Polish National Science Center ( NCN ) grants 2019/33/B/ST6/02988 and 2017/25/B/ST6/02553 , and by NSF Secure and Trustworthy Cyberspace Award 2131538 .
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/6/5

Y1 - 2022/6/5

N2 - In the Group Testing problem, the objective is to learn a subset K of some much larger domain N, using the shortest-possible sequence of queries Q. A feedback to a query provides some information about the intersection between the query and subset K. Several specific feedbacks have been studied in the literature, often proving different formulas for the estimate of the query complexity of the problem, defined as the shortest length of queries' sequence solving Group Testing problem with specific feedback. In this paper we study what are the properties of the feedback that influence the query complexity of Group Testing and what is their measurable impact. We propose a generic framework that covers a vast majority of relevant settings considered in the literature, which depends on two fundamental parameters of the feedback: input capacity α and output expressiveness β. They upper bound the logarithm of the size of the feedback function domain and image, respectively. To justify the value of the framework, we prove upper bounds on query complexity of non-adaptive, deterministic Group Testing under some “efficient” feedbacks, for minimum, maximum and general expressiveness, and complement them with a lower bound on all feedbacks with given parameters α,β. Our upper bounds also hold if the feedback function could get an input twisted by a malicious adversary, in case the intersection of a query and the hidden set is bigger than the feedback capacity α. We also show that slight change in the feedback function may result in substantial worsening of the query complexity. Additionally, we analyze explicitly constructed randomized counterparts of the deterministic results. Our results provide some insights to what are the most useful bits of information an output-restricted feedback could provide, and open a number of challenging research directions.

AB - In the Group Testing problem, the objective is to learn a subset K of some much larger domain N, using the shortest-possible sequence of queries Q. A feedback to a query provides some information about the intersection between the query and subset K. Several specific feedbacks have been studied in the literature, often proving different formulas for the estimate of the query complexity of the problem, defined as the shortest length of queries' sequence solving Group Testing problem with specific feedback. In this paper we study what are the properties of the feedback that influence the query complexity of Group Testing and what is their measurable impact. We propose a generic framework that covers a vast majority of relevant settings considered in the literature, which depends on two fundamental parameters of the feedback: input capacity α and output expressiveness β. They upper bound the logarithm of the size of the feedback function domain and image, respectively. To justify the value of the framework, we prove upper bounds on query complexity of non-adaptive, deterministic Group Testing under some “efficient” feedbacks, for minimum, maximum and general expressiveness, and complement them with a lower bound on all feedbacks with given parameters α,β. Our upper bounds also hold if the feedback function could get an input twisted by a malicious adversary, in case the intersection of a query and the hidden set is bigger than the feedback capacity α. We also show that slight change in the feedback function may result in substantial worsening of the query complexity. Additionally, we analyze explicitly constructed randomized counterparts of the deterministic results. Our results provide some insights to what are the most useful bits of information an output-restricted feedback could provide, and open a number of challenging research directions.

KW - Adversaries

KW - Deterministic algorithms

KW - Feedback functions

KW - Group testing

KW - Lower bound

KW - Non-adaptive algorithms

KW - Queries

KW - Randomized algorithms

UR - http://www.scopus.com/inward/record.url?scp=85127672608&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85127672608&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2022.03.026

DO - 10.1016/j.tcs.2022.03.026

M3 - Article

AN - SCOPUS:85127672608

VL - 919

SP - 18

EP - 35

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -