91 research outputs found
Model-Checking Problems as a Basis for Parameterized Intractability
Most parameterized complexity classes are defined in terms of a parameterized
version of the Boolean satisfiability problem (the so-called weighted
satisfiability problem). For example, Downey and Fellow's W-hierarchy is of
this form. But there are also classes, for example, the A-hierarchy, that are
more naturally characterised in terms of model-checking problems for certain
fragments of first-order logic.
Downey, Fellows, and Regan were the first to establish a connection between
the two formalisms by giving a characterisation of the W-hierarchy in terms of
first-order model-checking problems. We improve their result and then prove a
similar correspondence between weighted satisfiability and model-checking
problems for the A-hierarchy and the W^*-hierarchy. Thus we obtain very uniform
characterisations of many of the most important parameterized complexity
classes in both formalisms.
Our results can be used to give new, simple proofs of some of the core
results of structural parameterized complexity theory.Comment: Changes in since v2: Metadata update
On slicewise monotone parameterized problems and optimal proof systems for TAUT
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On p-optimal proof systems and logics for PTIME
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Consistency and optimality
Assume that the problem Qo is not solvable in polynomial time. For theories T containing a sufficiently rich part of true arithmetic we characterize T U {ConT} as the minimal extension of T proving for some algorithm that it decides Qo as fast as any algorithm B with the property that T proves that B decides Qo. Here, ConT claims the consistency of T. Moreover, we characterize problems with an optimal algorithm in terms of arithmetical theories
Lower bounds for kernelizations
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Hard instances of algorithms and proof systems
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Homomorphisms are a good basis for counting small subgraphs
We introduce graph motif parameters, a class of graph parameters that depend
only on the frequencies of constant-size induced subgraphs. Classical works by
Lov\'asz show that many interesting quantities have this form, including, for
fixed graphs , the number of -copies (induced or not) in an input graph
, and the number of homomorphisms from to .
Using the framework of graph motif parameters, we obtain faster algorithms
for counting subgraph copies of fixed graphs in host graphs : For graphs
on edges, we show how to count subgraph copies of in time
by a surprisingly simple algorithm. This
improves upon previously known running times, such as time
for -edge matchings or time for -cycles.
Furthermore, we prove a general complexity dichotomy for evaluating graph
motif parameters: Given a class of such parameters, we consider
the problem of evaluating on input graphs , parameterized
by the number of induced subgraphs that depends upon. For every recursively
enumerable class , we prove the above problem to be either FPT or
#W[1]-hard, with an explicit dichotomy criterion. This allows us to recover
known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms
in a uniform and simplified way, together with improved lower bounds.
Finally, we extend graph motif parameters to colored subgraphs and prove a
complexity trichotomy: For vertex-colored graphs and , where is from
a fixed class , we want to count color-preserving -copies in
. We show that this problem is either polynomial-time solvable or FPT or
#W[1]-hard, and that the FPT cases indeed need FPT time under reasonable
assumptions.Comment: An extended abstract of this paper appears at STOC 201
Strong isomorphism reductions in complexity theory
We give the first systematic study of strong isomorphism reductions, a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomorphim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a further type of reduction between classes of structures based on purely comparing for every n the number of nonisomorphic structures of cardinality at most n in both classes. Furthermore, in a more general setting we address the question of the existence of a maximal element in the partial ordering of the degrees
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