7,569 research outputs found
Algorithmic Meta-Theorems
Algorithmic meta-theorems are general algorithmic results applying to a whole
range of problems, rather than just to a single problem alone. They often have
a "logical" and a "structural" component, that is they are results of the form:
every computational problem that can be formalised in a given logic L can be
solved efficiently on every class C of structures satisfying certain
conditions. This paper gives a survey of algorithmic meta-theorems obtained in
recent years and the methods used to prove them. As many meta-theorems use
results from graph minor theory, we give a brief introduction to the theory
developed by Robertson and Seymour for their proof of the graph minor theorem
and state the main algorithmic consequences of this theory as far as they are
needed in the theory of algorithmic meta-theorems
Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited
Given a graph and two vertex sets satisfying a certain feasibility condition,
a reconfiguration problem asks whether we can reach one vertex set from the
other by repeating prescribed modification steps while maintaining feasibility.
In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic
meta-theorem for reconfiguration problems that says if the feasibility can be
expressed in monadic second-order logic (MSO), then the problem is
fixed-parameter tractable parameterized by , where
is the number of steps allowed to reach the target set. On the other
hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if is not
part of the parameter, then the problem is PSPACE-complete even on graphs of
bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case
where is not part of the parameter, using some structural graph
parameters incomparable with bandwidth. We show that if the feasibility is
defined in MSO, then the reconfiguration problem under the so-called token
jumping rule is fixed-parameter tractable parameterized by neighborhood
diversity. We also show that the problem is fixed-parameter tractable
parameterized by , where is the size of sets being
transformed. We finally complement the positive result for treedepth by showing
that the problem is PSPACE-complete on forests of depth .Comment: 25 pages, 2 figures, ESA 202
Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited
Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by treewidth + ?, where ? is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if ? is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case where ? is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by treedepth + k, where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3
Contraction Bidimensionality: the Accurate Picture
We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory -- the meta algorithmic framework to design efficient parameterized and approximation algorithms for contraction closed parameters
Definability Equals Recognizability for -Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph
property that can be defined by a sentence in counting monadic second order
logic (CMSOL) can be checked in linear time for graphs of bounded treewidth,
which is known as Courcelle's Theorem. These algorithms are constructed as
finite state tree automata, and hence every CMSOL-definable graph property is
recognizable. Courcelle also conjectured that the converse holds, i.e. every
recognizable graph property is definable in CMSOL for graphs of bounded
treewidth. We prove this conjecture for -outerplanar graphs, which are known
to have treewidth at most .Comment: 40 pages, 8 figure
Expanding the expressive power of Monadic Second-Order logic on restricted graph classes
We combine integer linear programming and recent advances in Monadic
Second-Order model checking to obtain two new algorithmic meta-theorems for
graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of
the well-known Monadic Second-Order logic by the addition of cardinality
constraints, can be solved in FPT time parameterized by vertex cover. The
second meta-theorem shows that the MSO partitioning problems introduced by Rao
can also be solved in FPT time with the same parameter. The significance of our
contribution stems from the fact that these formalisms can describe problems
which are W[1]-hard and even NP-hard on graphs of bounded tree-width.
Additionally, our algorithms have only an elementary dependence on the
parameter and formula. We also show that both results are easily extended from
vertex cover to neighborhood diversity.Comment: Accepted for IWOCA 201
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
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