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Theory Seminar: Testing Odd-Cycle-Freeness in Boolean Functions
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Arnab Bhattacharyya (MIT)
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Thursday, 30.6.2011, 12:30
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Room 337-8 Taub Bld.
Call a function f : {0,1}^n -> {0,1} odd-cycle-free if there are no x_1, ..., x_k in {0,1}^n with k an odd integer such that f(x_1) = ... = f(x_k) = 1 and x_1 + ... + x_k = 0. We show that one can distinguish odd-cycle-free functions from those eps-far from being odd-cycle-free by making poly(1/eps) queries to an evaluation oracle. To obtain this result, we use connections between Fourier analysis and spectral graph theory to show that one can reduce testing odd-cycle-freeness of Boolean functions to testing bipartiteness of dense graphs. Our results imply that one can estimate the smallest Fourier coefficient of a function upto additive error eps by making poly(1/eps) queries.

Our work forms part of a recent sequence of works that shows connections between testability of properties of Boolean functions and of graph properties. We also prove that there is a canonical tester for odd-cycle-freeness making poly(1/eps) queries, meaning that the testing algorithm operates by picking a random linear subspace of dimension O(log 1/eps) and then checking if the restriction of the function to the subspace is odd-cycle-free or not. The test is analyzed by studying the effect of random subspace restriction on the Fourier coefficients of a function. Our work implies that testing odd-cycle-freeness using a canonical tester instead of an arbitrary tester incurs no more than a polynomial blowup in the query complexity. The question of whether a canonical tester with polynomial blowup exists for all linear-invariant properties remains an open problem.

Joint work with Elena Grigorescu, Prasad Raghavendra, and Asaf Shapira.
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