On Symmetric Differences of NP-Hard Sets with Weakly P-Selective Sets

The symmetric differences of NP-hard sets with weakly-P-selective sets are investigated. We show that if there exist a weakly-P-selective set A and an NP-⩽Pm-hard set H such that H - AϵPbtt(sparse) and A — HϵPm(sparse) then P = NP. So no NP-⩽Pm-hard set has sparse symmetric difference with any weakly-P-selective set unless P = NP. The proof of our main result is an interesting application of the tree prunning techniques (Fortune 1979; Mahaney 1982). In addition, we show that there exists a P-selective set which has exponentially dense symmetric difference with every set in Pbtt(sparse).

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