100 research outputs found
Random strings and tt-degrees of Turing complete C.E. sets
We investigate the truth-table degrees of (co-)c.e.\ sets, in particular,
sets of random strings. It is known that the set of random strings with respect
to any universal prefix-free machine is Turing complete, but that truth-table
completeness depends on the choice of universal machine. We show that for such
sets of random strings, any finite set of their truth-table degrees do not meet
to the degree~0, even within the c.e. truth-table degrees, but when taking the
meet over all such truth-table degrees, the infinite meet is indeed~0. The
latter result proves a conjecture of Allender, Friedman and Gasarch. We also
show that there are two Turing complete c.e. sets whose truth-table degrees
form a minimal pair.Comment: 25 page
The complexity of computable categoricity
We show that the index set complexity of the computably categorical structures is View the MathML source-complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α , a computable structure that is computably categorical but not relatively View the MathML source-categorical
On problems without polynomial kernels (Extended abstract).
Abstract. Kernelization is a central technique used in parameterized algorithms, and in other techniques for coping with NP-hard problems. In this paper, we introduce a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexity-theoretic assumptions. These problems include kPath, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth or cliquewidth
Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem
The Workflow Satisfiability Problem (WSP) is a problem of practical interest
that arises whenever tasks need to be performed by authorized users, subject to
constraints defined by business rules. We are required to decide whether there
exists a plan -- an assignment of tasks to authorized users -- such that all
constraints are satisfied.
The WSP is, in fact, the conservative Constraint Satisfaction Problem (i.e.,
for each variable, here called task, we have a unary authorization constraint)
and is, thus, NP-complete. It was observed by Wang and Li (2010) that the
number k of tasks is often quite small and so can be used as a parameter, and
several subsequent works have studied the parameterized complexity of WSP
regarding parameter k.
We take a more detailed look at the kernelization complexity of WSP(\Gamma)
when \Gamma\ denotes a finite or infinite set of allowed constraints. Our main
result is a dichotomy for the case that all constraints in \Gamma\ are regular:
(1) We are able to reduce the number n of users to n' <= k. This entails a
kernelization to size poly(k) for finite \Gamma, and, under mild technical
conditions, to size poly(k+m) for infinite \Gamma, where m denotes the number
of constraints. (2) Already WSP(R) for some R \in \Gamma\ allows no polynomial
kernelization in k+m unless the polynomial hierarchy collapses.Comment: An extended abstract appears in the proceedings of IPEC 201
A join-based hybrid parameter for constraint satisfaction
We propose joinwidth, a new complexity parameter for the Constraint Satisfaction Problem (CSP). The definition of joinwidth is based on the arrangement of basic operations on relations (joins, projections, and pruning), which inherently reflects the steps required to solve the instance. We use joinwidth to obtain polynomial-time algorithms (if a corresponding decomposition is provided in the input) as well as fixed-parameter algorithms (if no such decomposition is provided) for solving the CSP.
Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account. It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw). Indeed, we show that any class of instances of bounded fhtw also has bounded joinwidth, and that there exist classes of instances of bounded joinwidth and unbounded fhtw, so bounded joinwidth properly generalizes bounded fhtw. We further show that bounded joinwidth also properly generalizes several other known hybrid restrictions, such as fhtw with degree constraints and functional dependencies. In this sense, bounded joinwidth can be seen as a unifying principle that explains the tractability of several seemingly unrelated classes of CSP instances
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