152 research outputs found
Fixed-parameter tractability, definability, and model checking
In this article, we study parameterized complexity theory from the
perspective of logic, or more specifically, descriptive complexity theory.
We propose to consider parameterized model-checking problems for various
fragments of first-order logic as generic parameterized problems and show how
this approach can be useful in studying both fixed-parameter tractability and
intractability. For example, we establish the equivalence between the
model-checking for existential first-order logic, the homomorphism problem for
relational structures, and the substructure isomorphism problem. Our main
tractability result shows that model-checking for first-order formulas is
fixed-parameter tractable when restricted to a class of input structures with
an excluded minor. On the intractability side, for every t >= 0 we prove an
equivalence between model-checking for first-order formulas with t quantifier
alternations and the parameterized halting problem for alternating Turing
machines with t alternations. We discuss the close connection between this
alternation hierarchy and Downey and Fellows' W-hierarchy.
On a more abstract level, we consider two forms of definability, called Fagin
definability and slicewise definability, that are appropriate for describing
parameterized problems. We give a characterization of the class FPT of all
fixed-parameter tractable problems in terms of slicewise definability in finite
variable least fixed-point logic, which is reminiscent of the Immerman-Vardi
Theorem characterizing the class PTIME in terms of definability in least
fixed-point logic.Comment: To appear in SIAM Journal on Computin
Definability of linear equation systems over groups and rings
Motivated by the quest for a logic for PTIME and recent insights that the
descriptive complexity of problems from linear algebra is a crucial aspect of
this problem, we study the solvability of linear equation systems over finite
groups and rings from the viewpoint of logical (inter-)definability. All
problems that we consider are decidable in polynomial time, but not expressible
in fixed-point logic with counting. They also provide natural candidates for a
separation of polynomial time from rank logics, which extend fixed-point logics
by operators for determining the rank of definable matrices and which are
sufficient for solvability problems over fields. Based on the structure theory
of finite rings, we establish logical reductions among various solvability
problems. Our results indicate that all solvability problems for linear
equation systems that separate fixed-point logic with counting from PTIME can
be reduced to solvability over commutative rings. Moreover, we prove closure
properties for classes of queries that reduce to solvability over rings, which
provides normal forms for logics extended with solvability operators. We
conclude by studying the extent to which fixed-point logic with counting can
express problems in linear algebra over finite commutative rings, generalising
known results on the logical definability of linear-algebraic problems over
finite fields
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
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension
is a complete isomorphism test for the class of all graphs of rank
width at most . Rank width is a graph invariant that, similarly to tree
width, measures the width of a certain style of hierarchical decomposition of
graphs; it is equivalent to clique width. It was known that isomorphism of
graphs of rank width is decidable in polynomial time (Grohe and Schweitzer,
FOCS 2015), but the best previously known algorithm has a running time
for a non-elementary function . Our result yields an isomorphism
test for graphs of rank width running in time . Another
consequence of our result is the first polynomial time canonisation algorithm
for graphs of bounded rank width. Our second main result is that fixed-point
logic with counting captures polynomial time on all graph classes of bounded
rank width.Comment: 32 page
Capturing Polynomial Time using Modular Decomposition
The question of whether there is a logic that captures polynomial time is one
of the main open problems in descriptive complexity theory and database theory.
In 2010 Grohe showed that fixed point logic with counting captures polynomial
time on all classes of graphs with excluded minors. We now consider classes of
graphs with excluded induced subgraphs. For such graph classes, an effective
graph decomposition, called modular decomposition, was introduced by Gallai in
1976. The graphs that are non-decomposable with respect to modular
decomposition are called prime. We present a tool, the Modular Decomposition
Theorem, that reduces (definable) canonization of a graph class C to
(definable) canonization of the class of prime graphs of C that are colored
with binary relations on a linearly ordered set. By an application of the
Modular Decomposition Theorem, we show that fixed point logic with counting
captures polynomial time on the class of permutation graphs. Within the proof
of the Modular Decomposition Theorem, we show that the modular decomposition of
a graph is definable in symmetric transitive closure logic with counting. We
obtain that the modular decomposition tree is computable in logarithmic space.
It follows that cograph recognition and cograph canonization is computable in
logarithmic space.Comment: 38 pages, 10 Figures. A preliminary version of this article appeared
in the Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer
Science (LICS '17
Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs
We show that the class of chordal claw-free graphs admits LREC-definable
canonization. LREC is a logic that extends first-order logic with counting
by an operator that allows it to formalize a limited form of recursion. This
operator can be evaluated in logarithmic space. It follows that there exists a
logarithmic-space canonization algorithm, and therefore a logarithmic-space
isomorphism test, for the class of chordal claw-free graphs. As a further
consequence, LREC captures logarithmic space on this graph class. Since
LREC is contained in fixed-point logic with counting, we also obtain that
fixed-point logic with counting captures polynomial time on the class of
chordal claw-free graphs.Comment: 34 pages, 13 figure
On the Parameterized Intractability of Monadic Second-Order Logic
One of Courcelle's celebrated results states that if C is a class of graphs
of bounded tree-width, then model-checking for monadic second order logic
(MSO_2) is fixed-parameter tractable (fpt) on C by linear time parameterized
algorithms, where the parameter is the tree-width plus the size of the formula.
An immediate question is whether this is best possible or whether the result
can be extended to classes of unbounded tree-width. In this paper we show that
in terms of tree-width, the theorem cannot be extended much further. More
specifically, we show that if C is a class of graphs which is closed under
colourings and satisfies certain constructibility conditions and is such that
the tree-width of C is not bounded by \log^{84} n then MSO_2-model checking is
not fpt unless SAT can be solved in sub-exponential time. If the tree-width of
C is not poly-logarithmically bounded, then MSO_2-model checking is not fpt
unless all problems in the polynomial-time hierarchy can be solved in
sub-exponential time
Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs
We show that the class of chordal claw-free graphs admits LREC=-definable canonization. LREC= is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm for the class of chordal claw-free graphs, and that LREC= captures logarithmic space on this graph class. Since LREC= is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs
Canonizing Graphs of Bounded Tree Width in Logspace
Graph canonization is the problem of computing a unique representative, a
canon, from the isomorphism class of a given graph. This implies that two
graphs are isomorphic exactly if their canons are equal. We show that graphs of
bounded tree width can be canonized by logarithmic-space (logspace) algorithms.
This implies that the isomorphism problem for graphs of bounded tree width can
be decided in logspace. In the light of isomorphism for trees being hard for
the complexity class logspace, this makes the ubiquitous class of graphs of
bounded tree width one of the few classes of graphs for which the complexity of
the isomorphism problem has been exactly determined.Comment: 26 page
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