2,875 research outputs found
Parameterized Complexity of Graph Constraint Logic
Graph constraint logic is a framework introduced by Hearn and Demaine, which
provides several problems that are often a convenient starting point for
reductions. We study the parameterized complexity of Constraint Graph
Satisfiability and both bounded and unbounded versions of Nondeterministic
Constraint Logic (NCL) with respect to solution length, treewidth and maximum
degree of the underlying constraint graph as parameters. As a main result we
show that restricted NCL remains PSPACE-complete on graphs of bounded
bandwidth, strengthening Hearn and Demaine's framework. This allows us to
improve upon existing results obtained by reduction from NCL. We show that
reconfiguration versions of several classical graph problems (including
independent set, feedback vertex set and dominating set) are PSPACE-complete on
planar graphs of bounded bandwidth and that Rush Hour, generalized to boards, is PSPACE-complete even when is at most a constant
Implicit function theorem over free groups
We introduce the notion of a regular quadratic equation and a regular NTQ
system over a free group. We prove the results that can be described as
Implicit function theorems for algebraic varieties corresponding to regular
quadratic and NTQ systems. We will also show that the Implicit function theorem
is true only for these varieties. In algebraic geometry such results would be
described as lifting solutions of equations into generic points. From the model
theoretic view-point we claim the existence of simple Skolem functions for
particular -formulas over free groups. Proving these theorems
we describe in details a new version of the Makanin-Razborov process for
solving equations in free groups. We also prove a weak version of the Implicit
function theorem for NTQ systems which is one of the key results in the
solution of the Tarski's problems about the elementary theory of a free group.Comment: 144 pages, 16 figure
Manifolds with non-stable fundamental groups at infinity, II
In this paper we continue an earlier study of ends non-compact manifolds. The
over-arching goal is to investigate and obtain generalizations of Siebenmann's
famous collaring theorem that may be applied to manifolds having non-stable
fundamental group systems at infinity. In this paper we show that, for
manifolds with compact boundary, the condition of inward tameness has
substatial implications for the algebraic topology at infinity. In particular,
every inward tame manifold with compact boundary has stable homology (in all
dimensions) and semistable fundamental group at each of its ends. In contrast,
we also construct examples of this sort which fail to have perfectly semistable
fundamental group at infinity. In doing so, we exhibit the first known examples
of open manifolds that are inward tame and have vanishing Wall finiteness
obstruction at infinity, but are not pseudo-collarable.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper7.abs.htm
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