computational complexity

, Volume 21, Issue 4, pp 643–670 | Cite as

On the power of unambiguity in log-space

  • A. Pavan
  • Raghunath TewariEmail author
  • N. V. Vinodchandran


We report progress on the NL versus UL problem.
  • We show that counting the number of s-t paths in graphs where the number of s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class. Several new upper bounds follow from this including \({{{ReachFewL} \subseteq {UL}}}\) and \({{{LFew} \subseteq {UL}^{FewL}}}\)

  • We investigate the complexity of min-uniqueness—a central notion in studying the NL versus UL problem. In this regard we revisit the class OptL[log n] and introduce UOptL[log n], an unambiguous version of OptL[log n]. We investigate the relation between UOptL[log n] and other existing complexity classes.

  • We consider the unambiguous hierarchies over UL and UOptL[log n]. We show that the hierarchy over UOptL[log n] collapses. This implies that \({{{ULH} \subseteq {L}^{{promiseUL}}}}\) thus collapsing the UL hierarchy.

  • We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages, which is log-space equivalent to the reachability problem in planar graphs and hence is in UL.


Log-space complexity unambiguous computations graph reachability log-space optimization hardness 

Subject classification

68Q05 68Q10 68Q15 68Q17 


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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • A. Pavan
    • 1
  • Raghunath Tewari
    • 2
    Email author
  • N. V. Vinodchandran
    • 3
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA
  2. 2.Department of Computer Science and EngineeringIndian Institute of Technology KanpurKanpurIndia
  3. 3.Department of Computer Science and EngineeringUniversity of Nebraska–LincolnLincolnUSA

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