Derandomizing Graph Tests for Homomorphism

Abstract

In this article, we study the randomness-efficient graph tests for homomorphism over arbitrary groups (which can be used in locally testing the Hadamard code and PCP construction). We try to optimize both the amortized-tradeoff (between number of queries and error probability) and the randomness complexity of the homomorphism test simultaneously. For an abelian group $G=\mathbb Z_{p}^{m}$, by using the λ-biased set S of G, we show that, on any given bipartite graph H = (V ₁,V ₂;E), the graph test for linearity over G is a test with randomness complexity |V ₁|log|G| + |V ₂|O(log|S|), query complexity |V ₁| + |V ₂| + |E| and error probability at most p ^− |E| + (1 − p ^− |E|)·δ for any f which is $1-p^{-1}(1+\frac{\sqrt{\delta^{2}-\lambda}}{2})$-far from being affine linear. For a non-abelian group G, we introduce a random walk of some length, ℓ say, on expander graphs to design a probabilistic homomorphism test over G with randomness complexity log|G| + O(loglog|G|), query complexity 2ℓ + 1 and error probability at most $1- \frac{\displaystyle\delta^{2}\ell^{2}}{\displaystyle(\delta\ell+\delta^{2}\ell^{2}- \delta^{2}\ell)+2\psi(\lambda,\ell)}$ for any f which is 2δ/(1 − λ)-far from being affine homomorphism, here $\psi(\lambda,\ell)=\sum_{0\leq i<j\leq\ell-1}\lambda^{j-i-1}$.