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 𝖯 ≠ NP, 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.
@InProceedings{blanc_et_al:LIPIcs.ITCS.2023.18, author = {Blanc, Guy and Koch, Caleb and Lange, Jane and Strassle, Carmen and Tan, Li-Yang}, title = {{Certification with an NP Oracle}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {18:1--18:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.18}, URN = {urn:nbn:de:0030-drops-175217}, doi = {10.4230/LIPIcs.ITCS.2023.18}, annote = {Keywords: Certificate complexity, Boolean functions, polynomial hierarchy, hardness of approximation} }
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