253 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
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
Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs
We show that the model-checking problem for successor-invariant first-order
logic is fixed-parameter tractable on graphs with excluded topological
subgraphs when parameterised by both the size of the input formula and the size
of the exluded topological subgraph. Furthermore, we show that model-checking
for order-invariant first-order logic is tractable on coloured posets of
bounded width, parameterised by both the size of the input formula and the
width of the poset.
Our result for successor-invariant FO extends previous results for this logic
on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors
(Eickmeyer et al., LICS 2013), further narrowing the gap between what is known
for FO and what is known for successor-invariant FO. The proof uses Grohe and
Marx's structure theorem for graphs with excluded topological subgraphs. For
order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO
carries over to order-invariant FO
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