3 research outputs found

    Structure of computations in parallel complexity classes

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    Issued as Annual report, and Final project report, Project no. G-36-67

    On space efficiency of algorithms working on structural decompositions of graphs

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    Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition's width. Following the work of Allender et al. [Theory of Computing, '14], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we complete the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition's depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new version is augmented with a space-efficient algorithm for Dominating Set using the Chinese remainder theore

    The expressibility of nondeterministic auxiliary stack automata and its relation to treesize bounded alternating auxiliary pushdown automata

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    Two aspects of nondeterministic auxiliary stack automata (NAuxSA) are studied. The first is the expressibility of NAuxSA. More specifically, it is shown that the polynomial hierarchy can be characterized in terms of AuxSA with resource bounds. The second aspect is a duality relation between NAuxSA and alternating auxiliary pushdown automata (Alt-AuxPDA) connecting time bounds on the former with treesize bounds on the latter
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