18,979 research outputs found
Completeness Results for Parameterized Space Classes
The parameterized complexity of a problem is considered "settled" once it has
been shown to lie in FPT or to be complete for a class in the W-hierarchy or a
similar parameterized hierarchy. Several natural parameterized problems have,
however, resisted such a classification. At least in some cases, the reason is
that upper and lower bounds for their parameterized space complexity have
recently been obtained that rule out completeness results for parameterized
time classes. In this paper, we make progress in this direction by proving that
the associative generability problem and the longest common subsequence problem
are complete for parameterized space classes. These classes are defined in
terms of different forms of bounded nondeterminism and in terms of simultaneous
time--space bounds. As a technical tool we introduce a "union operation" that
translates between problems complete for classical complexity classes and for
W-classes.Comment: IPEC 201
Hairdressing in groups: a survey of combings and formal languages
A group is combable if it can be represented by a language of words
satisfying a fellow traveller property; an automatic group has a synchronous
combing which is a regular language. This article surveys results for combable
groups, in particular in the case where the combing is a formal language.Comment: 17 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper24.abs.htm
Multipass automata and group word problems
We introduce the notion of multipass automata as a generalization of pushdown
automata and study the classes of languages accepted by such machines. The
class of languages accepted by deterministic multipass automata is exactly the
Boolean closure of the class of deterministic context-free languages while the
class of languages accepted by nondeterministic multipass automata is exactly
the class of poly-context-free languages, that is, languages which are the
intersection of finitely many context-free languages. We illustrate the use of
these automata by studying groups whose word problems are in the above classes
On space efficiency of algorithms working on structural decompositions of graphs
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
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