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We study the problem of certification: given queries to a function f : {0,1}n → {0,1} with certificate complexity ≤ k and an input x⋆, output a size-k certificate for f’s value on x⋆. For monotone functions, a classic local search algorithm of Angluin accomplishes this task with n queries, which we show is optimal for local search algorithms. Our main result is a new algorithm for certifying monotone functions with O(k8 logn) queries, which comes close to matching the information-theoretic lower bound of Ω(k logn). The design and analysis of our algorithm are based on a new connection to threshold phenomena in monotone functions. We further prove exponential-in-k lower bounds when f is non-monotone, and when f is monotone but the algorithm is only given random examples of f. These lower bounds show that assumptions on the structure of f and query access to it are both necessary for the polynomial dependence on k that we achieve.
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Certification with an NP oracle
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.
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- Award ID(s):
- 2006664
- PAR ID:
- 10434685
- Date Published:
- Journal Name:
- Proceedings of the 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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