Certification with an NP Oracle
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In the certification problem, the algorithm is given a function f with certificate complexity k and an input x^⋆, and the goal is to find a certificate of size ≤poly(k) for f's value at x^⋆. This problem is in 𝖭𝖯^𝖭𝖯, and assuming 𝖯𝖭𝖯, is not in 𝖯. Prior works, dating back to Valiant in 1984, have therefore sought to design efficient algorithms by imposing assumptions on f such as monotonicity. Our first result is a 𝖡𝖯𝖯^𝖭𝖯 algorithm for the general problem. The key ingredient is a new notion of the balanced influence of variables, a natural variant of influence that corrects for the bias of the function. Balanced influences can be accurately estimated via uniform generation, and classic 𝖡𝖯𝖯^𝖭𝖯 algorithms are known for the latter task. We then consider certification with stricter instance-wise guarantees: for each x^⋆, find a certificate whose size scales with that of the smallest certificate for x^⋆. In sharp contrast with our first result, we show that this problem is 𝖭𝖯^𝖭𝖯-hard even to approximate. We obtain an optimal inapproximability ratio, adding to a small handful of problems in the higher levels of the polynomial hierarchy for which optimal inapproximability is known. Our proof involves the novel use of bit-fixing dispersers for gap amplification.
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