Constant Degree Direct Product Testers with Small Soundness
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910529912569856 |
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| author | Bafna, Mitali Lifshitz, Noam Minzer, Dor |
| author_facet | Bafna, Mitali Lifshitz, Noam Minzer, Dor |
| contents | Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(σ) = (f(σ_1), \ldots, f(σ_k))$ for each $k$-face $σ$. In an effort to simplify components of the PCP theorem, Goldreich and Safra introduced the problem of direct product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$ is correlated with a direct product function by querying $F$ on only $2$ inputs. Dinur and Kaufman conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all $δ>0$, there exists a family of high-dimensional expanders with degree $O_δ(1)$ and a $2$-query direct product tester with soundness $δ$.
We use the characterization given by a subset of the authors and independently by Dikstein and Dinur, who showed that some form of non-Abelian coboundary expansion (which they called "Unique-Games coboundary expansion") is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by Chapman and Lubotzky satisfies the necessary conditions, thus admitting a 2-query direct product tester with small soundness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00850 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constant Degree Direct Product Testers with Small Soundness Bafna, Mitali Lifshitz, Noam Minzer, Dor Computational Complexity Combinatorics Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(σ) = (f(σ_1), \ldots, f(σ_k))$ for each $k$-face $σ$. In an effort to simplify components of the PCP theorem, Goldreich and Safra introduced the problem of direct product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$ is correlated with a direct product function by querying $F$ on only $2$ inputs. Dinur and Kaufman conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all $δ>0$, there exists a family of high-dimensional expanders with degree $O_δ(1)$ and a $2$-query direct product tester with soundness $δ$. We use the characterization given by a subset of the authors and independently by Dikstein and Dinur, who showed that some form of non-Abelian coboundary expansion (which they called "Unique-Games coboundary expansion") is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by Chapman and Lubotzky satisfies the necessary conditions, thus admitting a 2-query direct product tester with small soundness. |
| title | Constant Degree Direct Product Testers with Small Soundness |
| topic | Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2402.00850 |