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 , 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 -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 for any f which is 2δ/(1-λ)-far from being affine homomorphism, here . © 2008 Springer-Verlag Berlin Heidelberg.