Abstract
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. [5], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [18], we prove that the question is closely related to a conjecture that the L ongest C ommon S ubsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we extend 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.
Highlights
Treewidth is a parameter that measures how a graph can be decomposed into a tree-like structure, called a tree decomposition
Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 57:2 On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs
Mirroring Theorem 1, we prove that computations on tree-depth decompositions exactly correspond to the class NAuxSA[poly, log, s ]: problems that can be time space height decided by a non-deterministic Turing Machine that uses polynomial time and logarithmic space, and has access to an auxiliary stack of maximum height s
Summary
Treewidth is a parameter that measures how a graph can be decomposed into a tree-like structure, called a tree decomposition. The best known determinization for N[poly, s ] come from a brute-force approach or Savitch’s theorem [43], yielding retime space spectively (for s(n) ≥ lg n) D[2O(s)] = D[2O(s), 2O(s)] and D[s · log ] = D[2O(s·log n), s · log ] In this manner, Allender et al conclude that, intuitively speaking, achieving better time-space tradeoffs for algorithms working on path and tree decompositions of small width would require developing a general technique of improving upon the tradeoff of Savitch. Mirroring Theorem 1, we prove that computations on tree-depth decompositions exactly correspond to the class NAuxSA[poly, log , s ]: problems that can be time space height decided by a non-deterministic Turing Machine that uses polynomial time and logarithmic space, and has access to an auxiliary stack of maximum height s. 57:6 On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs
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