2026-07-21T14:12:59Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:14102 2024-03-06T10:53:28Z ddc:004 open_access

Direct Sum and Partitionability Testing over General Groups Bogdanov, Andrej Prakriya, Gautam Direct Sum Test Function Partitionability Hypercontractive Inequality Property Testing A function f(x₁, … , x_n) from a product domain 𝒟₁ × ⋯ × 𝒟_n to an abelian group 𝒢 is a direct sum if it is of the form f₁(x₁) + ⋯ + f_n(x_n). We present a new 4-query direct sum test with optimal (up to constant factors) soundness error. This generalizes a result of Dinur and Golubev (RANDOM 2019) which is tailored to the target group 𝒢 = ℤ₂. As a special case, we obtain an optimal affinity test for 𝒢-valued functions on domain {0, 1}ⁿ under product measure. Our analysis relies on the hypercontractivity of the binary erasure channel. We also study the testability of function partitionability over product domains into disjoint components. A 𝒢-valued f(x₁, … , x_n) is k-direct sum partitionable if it can be written as a sum of functions over k nonempty disjoint sets of inputs. A function f(x₁, … , x_n) with unstructured product range ℛ^k is direct product partitionable if its outputs depend on disjoint sets of inputs. We show that direct sum partitionability and direct product partitionability are one-sided error testable with O((n - k)(log n + 1/ε) + 1/ε) adaptive queries and O((n/ε) log²(n/ε)) nonadaptive queries, respectively. Both bounds are tight up to the logarithmic factors for constant ε even with respect to adaptive, two-sided error testers. We also give a non-adaptive one-sided error tester for direct sum partitionability with query complexity O(kn² (log n)² / ε). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Andrej Bogdanov and Gautam Prakriya 2021 Is Part Of LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2021.33 urn:nbn:de:0030-drops-141028 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.33 eng https://creativecommons.org/licenses/by/4.0/legalcode