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On the power of unambiguity in log-space

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Abstract

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.

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Authors and Affiliations

  1. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    A. Pavan

  2. Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, India

    Raghunath Tewari

  3. Department of Computer Science and Engineering, University of Nebraska–Lincoln, Lincoln, NE, 68588, USA

    N. V. Vinodchandran

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  1. A. Pavan
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  2. Raghunath Tewari
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  3. N. V. Vinodchandran
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Correspondence to Raghunath Tewari.

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Pavan, A., Tewari, R. & Vinodchandran, N.V. On the power of unambiguity in log-space. comput. complex. 21, 643–670 (2012). https://doi.org/10.1007/s00037-012-0047-3

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