27 research outputs found
The Complexity of Rooted Phylogeny Problems
Several computational problems in phylogenetic reconstruction can be
formulated as restrictions of the following general problem: given a formula in
conjunctive normal form where the literals are rooted triples, is there a
rooted binary tree that satisfies the formula? If the formulas do not contain
disjunctions, the problem becomes the famous rooted triple consistency problem,
which can be solved in polynomial time by an algorithm of Aho, Sagiv,
Szymanski, and Ullman. If the clauses in the formulas are restricted to
disjunctions of negated triples, Ng, Steel, and Wormald showed that the problem
remains NP-complete. We systematically study the computational complexity of
the problem for all such restrictions of the clauses in the input formula. For
certain restricted disjunctions of triples we present an algorithm that has
sub-quadratic running time and is asymptotically as fast as the fastest known
algorithm for the rooted triple consistency problem. We also show that any
restriction of the general rooted phylogeny problem that does not fall into our
tractable class is NP-complete, using known results about the complexity of
Boolean constraint satisfaction problems. Finally, we present a pebble game
argument that shows that the rooted triple consistency problem (and also all
generalizations studied in this paper) cannot be solved by Datalog
The complexity of rooted phylogeny problems
ABSTRACT Several computational problems in phylogenetic reconstruction can be formulated as restrictions of the following general problem: given a formula in conjunctive normal form where the atomic formulas are rooted triples, is there a rooted binary tree that satisfies the formula? If the formulas do not contain disjunctions and negations, the problem becomes the famous rooted triple consistency problem, which can be solved in polynomial time by an algorithm of Aho, Sagiv, Szymanski, and Ullman. If the clauses in the formulas are restricted to disjunctions of negated triples, Ng, Steel, and Wormald showed that the problem remains NP-complete. We systematically study the computational complexity of the problem for all such restrictions of the clauses in the input formula. For certain restricted disjunctions of triples we present an algorithm that has sub-quadratic running time and is asymptotically as fast as the fastest known algorithm for the rooted triple consistency problem. We also show that any restriction of the general rooted phylogeny problem that does not fall into our tractable class is NP-complete, using known results about the complexity of Boolean constraint satisfaction problems. Finally, we present a pebble game argument that shows that the rooted triple consistency problem (and also all generalizations studied in this paper) cannot be solved by Datalog
On the Descriptive Complexity of Temporal Constraint Satisfaction Problems
Finite-domain constraint satisfaction problems are either solvable by
Datalog, or not even expressible in fixed-point logic with counting. The border
between the two regimes coincides with an important dichotomy in universal
algebra; in particular, the border can be described by a strong height-one
Maltsev condition. For infinite-domain CSPs, the situation is more complicated
even if the template structure of the CSP is model-theoretically tame. We prove
that there is no Maltsev condition that characterizes Datalog already for the
CSPs of first-order reducts of (Q;<); such CSPs are called temporal CSPs and
are of fundamental importance in infinite-domain constraint satisfaction. Our
main result is a complete classification of temporal CSPs that can be expressed
in one of the following logical formalisms: Datalog, fixed-point logic (with or
without counting), or fixed-point logic with the Boolean rank operator. The
classification shows that many of the equivalent conditions in the finite fail
to capture expressibility in Datalog or fixed-point logic already for temporal
CSPs.Comment: 57 page
Complexity Classification Transfer for CSPs via Algebraic Products
We study the complexity of infinite-domain constraint satisfaction problems:
our basic setting is that a complexity classification for the CSPs of
first-order expansions of a structure can be transferred to a
classification of the CSPs of first-order expansions of another structure
. We exploit a product of structures (the algebraic product) that
corresponds to the product of the respective polymorphism clones and present a
complete complexity classification of the CSPs for first-order expansions of
the -fold algebraic power of . This is proved by various
algebraic and logical methods in combination with knowledge of the
polymorphisms of the tractable first-order expansions of and
explicit descriptions of the expressible relations in terms of syntactically
restricted first-order formulas. By combining our classification result with
general classification transfer techniques, we obtain surprisingly strong new
classification results for highly relevant formalisms such as Allen's Interval
Algebra, the -dimensional Block Algebra, and the Cardinal Direction
Calculus, even if higher-arity relations are allowed. Our results confirm the
infinite-domain tractability conjecture for classes of structures that have
been difficult to analyse with older methods. For the special case of
structures with binary signatures, the results can be substantially
strengthened and tightly connected to Ord-Horn formulas; this solves several
longstanding open problems from the AI literature.Comment: 61 pages, 1 figur
Schaefer's theorem for graphs
Schaefer's theorem is a complexity classification result for so-called
Boolean constraint satisfaction problems: it states that every Boolean
constraint satisfaction problem is either contained in one out of six classes
and can be solved in polynomial time, or is NP-complete.
We present an analog of this dichotomy result for the propositional logic of
graphs instead of Boolean logic. In this generalization of Schaefer's result,
the input consists of a set W of variables and a conjunction \Phi\ of
statements ("constraints") about these variables in the language of graphs,
where each statement is taken from a fixed finite set \Psi\ of allowed
quantifier-free first-order formulas; the question is whether \Phi\ is
satisfiable in a graph.
We prove that either \Psi\ is contained in one out of 17 classes of graph
formulas and the corresponding problem can be solved in polynomial time, or the
problem is NP-complete. This is achieved by a universal-algebraic approach,
which in turn allows us to use structural Ramsey theory. To apply the
universal-algebraic approach, we formulate the computational problems under
consideration as constraint satisfaction problems (CSPs) whose templates are
first-order definable in the countably infinite random graph. Our method to
classify the computational complexity of those CSPs is based on a
Ramsey-theoretic analysis of functions acting on the random graph, and we
develop general tools suitable for such an analysis which are of independent
mathematical interest.Comment: 54 page
Tractable Ontology-Mediated Query Answering with Datatypes
Adding datatypes to ontology-mediated queries (OMQs) often makes query answering hard, even for lightweight languages. As a consequence, the use of datatypes in ontologies, e.g. in OWL 2 QL, has been severely restricted. We propose a new, non-uniform, way of analyzing the data-complexity of OMQ answering with datatypes. Instead of restricting the ontology language we aim at a classification of the patterns of datatype atoms in OMQs into those that can occur in non-tractable OMQs and those that only occur in tractable OMQs. To this end we establish a close link between OMQ answering with datatypes and constraint satisfaction problems (CSPs) over the datatypes. Given that query answering in this setting is undecidable in general already for very simple datatypes, we introduce, borrowing from the database literature, a property of OMQs called the Bounded Match Depth Property (BMDP). We apply the link to CSPs– using results and techniques in universal algebra and model theory–to prove PTIME/co-NP dichotomies for OMQs with the BDMP over Horn-ALCHI extended with (1) all finite datatypes, (2) rational numbers with linear order and (3) certain families of datatypes over the integers with the successor relation