White-box vs. black-box complexity of search problems: Ramsey and graph property testing

Ilan Komargodski*, Moni Naor, Eylon Yogev

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

21 Scopus citations

Abstract

Ramsey theory assures us that in any graph there is a clique or independent set of a certain size, roughly logarithmic in the graph size. But how difficult is it to find the clique or independent set? If the graph is given explicitly, then it is possible to do so while examining a linear number of edges. If the graph is given by a black-box, where to figure out whether a certain edge exists the box should be queried, then a large number of queries must be issued. But what if one is given a program or circuit for computing the existence of an edge? This problem was raised by Buss and Goldberg and Papadimitriou in the context of TFNP, search problems with a guaranteed solution.We examine the relationship between black-box complexity and white-box complexity for search problems with guaranteed solution such as the above Ramsey problem. We show that under the assumption that collision resistant hash function exist (which follows from the hardness of problems such as factoring, discrete-log and learning with errors) the white-box Ramsey problem is hard and this is true even if one is looking for a much smaller clique or independent set than the theorem guarantees.In general, one cannot hope to translate all black-box hardness for TFNP into white-box hardness: we show this by adapting results concerning the random oracle methodology and the impossibility of instantiating it.Another model we consider is the succinct black-box, where there is a known upper bound on the size of the black-box (but no limit on the computation time). In this case we show that for all TFNP problems there is an upper bound on the number of queries proportional to the description size of the box times the solution size. On the other hand, for promise problems this is not the case.Finally, we consider the complexity of graph property testing in the white-box model. We show a property which is hard to test even when one is given the program for computing the graph. The hard property is whether the graph is a two-source extractor.

Original languageAmerican English
Title of host publicationProceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
PublisherIEEE Computer Society
Pages622-632
Number of pages11
ISBN (Electronic)9781538634646
DOIs
StatePublished - 10 Nov 2017
Externally publishedYes
Event58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 - Berkeley, United States
Duration: 15 Oct 201717 Oct 2017

Publication series

NameAnnual Symposium on Foundations of Computer Science - Proceedings
Volume2017-October
ISSN (Print)0272-5428

Conference

Conference58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
Country/TerritoryUnited States
CityBerkeley
Period15/10/1717/10/17

Bibliographical note

Publisher Copyright:
© 2017 IEEE.

Keywords

  • Search problems
  • black-box complexity
  • collision-resistant hashing
  • the Ramsey problem
  • white-box complexity

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