computational complexity

, Volume 22, Issue 2, pp 345–383 | Cite as

Is Valiant–Vazirani’s isolation probability improvable?

  • Holger DellEmail author
  • Valentine Kabanets
  • Dieter van Melkebeek
  • Osamu Watanabe


The Isolation Lemma of Valiant and Vazirani (Theor Comput Sci 47:85–93, 1986) provides an efficient procedure for isolating a satisfying assignment of a given satisfiable circuit: Given a Boolean circuit C on n input variables, the procedure outputs a new circuit C′ on the same n input variables such that (i) every satisfying assignment of C′ also satisfies C and (ii) if C is satisfiable, then C′ has exactly one satisfying assignment. In particular, if C is unsatisfiable, then (i) implies that C′ is unsatisfiable. The Valiant–Vazirani procedure is randomized, and when C is satisfiable, it produces a uniquely satisfiable circuit C′ with probability Ω(1/n).

Is it possible to have an efficient deterministic witness-isolating procedure? Or, at least, is it possible to improve the success probability of a randomized procedure to a large constant? We prove that there exists a non-uniform randomized polynomial-time witness-isolating procedure with success probability bigger than 2/3 if and only if NP \({\subseteq }\) P/poly. We establish similar results for other variants of witness isolation, such as reductions that remove all but an odd number of satisfying assignments of a satisfiable circuit.

We also consider a blackbox setting of witness isolation that generalizes the setting of the Valiant–Vazirani Isolation Lemma and give an upper bound of O(1/n) on the success probability for a natural class of randomized witness-isolating procedures.


Isolation Lemma unique satisfiability parity satisfiability derandomization 

Subject Classification

68Q15 68Q17 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Holger Dell
    • 1
    Email author
  • Valentine Kabanets
    • 2
  • Dieter van Melkebeek
    • 1
  • Osamu Watanabe
    • 3
  1. 1.Department of Computer SciencesUniversity of WisconsinMadisonUSA
  2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  3. 3.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan

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