25,451 research outputs found
Approximation for Maximum Surjective Constraint Satisfaction Problems
Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are
computational problems where we are given a set of variables denoting values
from a finite domain B and a set of constraints on the variables. A solution to
such a problem is a surjective mapping from the set of variables to B such that
the number of satisfied constraints is maximized. We study the approximation
performance that can be acccchieved by algorithms for these problems, mainly by
investigating their relation with Max-CSPs (which are the corresponding
problems without the surjectivity requirement). Our work gives a complexity
dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the
assumption that there is a complexity dichotomy for Max-CSP(B) between PO and
APX-complete, which has already been proved on the Boolean domain and 3-element
domains
Solving Set Constraint Satisfaction Problems using ROBDDs
In this paper we present a new approach to modeling finite set domain
constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We
show that it is possible to construct an efficient set domain propagator which
compactly represents many set domains and set constraints using ROBDDs. We
demonstrate that the ROBDD-based approach provides unprecedented flexibility in
modeling constraint satisfaction problems, leading to performance improvements.
We also show that the ROBDD-based modeling approach can be extended to the
modeling of integer and multiset constraint problems in a straightforward
manner. Since domain propagation is not always practical, we also show how to
incorporate less strict consistency notions into the ROBDD framework, such as
set bounds, cardinality bounds and lexicographic bounds consistency. Finally,
we present experimental results that demonstrate the ROBDD-based solver
performs better than various more conventional constraint solvers on several
standard set constraint problems
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
Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems
A complete classification of the computational complexity of the fixed-point
existence problem for boolean dynamical systems, i.e., finite discrete
dynamical systems over the domain {0, 1}, is presented. For function classes F
and graph classes G, an (F, G)-system is a boolean dynamical system such that
all local transition functions lie in F and the underlying graph lies in G. Let
F be a class of boolean functions which is closed under composition and let G
be a class of graphs which is closed under taking minors. The following
dichotomy theorems are shown: (1) If F contains the self-dual functions and G
contains the planar graphs then the fixed-point existence problem for (F,
G)-systems with local transition function given by truth-tables is NP-complete;
otherwise, it is decidable in polynomial time. (2) If F contains the self-dual
functions and G contains the graphs having vertex covers of size one then the
fixed-point existence problem for (F, G)-systems with local transition function
given by formulas or circuits is NP-complete; otherwise, it is decidable in
polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24
versio
Existentially Restricted Quantified Constraint Satisfaction
The quantified constraint satisfaction problem (QCSP) is a powerful framework
for modelling computational problems. The general intractability of the QCSP
has motivated the pursuit of restricted cases that avoid its maximal
complexity. In this paper, we introduce and study a new model for investigating
QCSP complexity in which the types of constraints given by the existentially
quantified variables, is restricted. Our primary technical contribution is the
development and application of a general technology for proving positive
results on parameterizations of the model, of inclusion in the complexity class
coNP
Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems
We present dichotomy theorems regarding the computational complexity of
counting fixed points in boolean (discrete) dynamical systems, i.e., finite
discrete dynamical systems over the domain {0,1}. For a class F of boolean
functions and a class G of graphs, an (F,G)-system is a boolean dynamical
system with local transitions functions lying in F and graphs in G. We show
that, if local transition functions are given by lookup tables, then the
following complexity classification holds: Let F be a class of boolean
functions closed under superposition and let G be a graph class closed under
taking minors. If F contains all min-functions, all max-functions, or all
self-dual and monotone functions, and G contains all planar graphs, then it is
#P-complete to compute the number of fixed points in an (F,G)-system; otherwise
it is computable in polynomial time. We also prove a dichotomy theorem for the
case that local transition functions are given by formulas (over logical
bases). This theorem has a significantly more complicated structure than the
theorem for lookup tables. A corresponding theorem for boolean circuits
coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on
Theoretical Computer Science (ICTCS'2007
The complexity of conservative finite-valued CSPs
We study the complexity of valued constraint satisfaction problems (VCSP). A
problem from VCSP is characterised by a \emph{constraint language}, a fixed set
of cost functions over a finite domain. An instance of the problem is specified
by a sum of cost functions from the language and the goal is to minimise the
sum. We consider the case of so-called \emph{conservative} languages; that is,
languages containing all unary cost functions, thus allowing arbitrary
restrictions on the domains of the variables. This problem has been studied by
Bulatov [LICS'03] for -valued languages (i.e. CSP), by
Cohen~\etal\ (AIJ'06) for Boolean domains, by Deineko et al. (JACM'08) for
-valued cost functions (i.e. Max-CSP), and by Takhanov (STACS'10) for
-valued languages containing all finite-valued unary cost
functions (i.e. Min-Cost-Hom).
We give an elementary proof of a complete complexity classification of
conservative finite-valued languages: we show that every conservative
finite-valued language is either tractable or NP-hard. This is the \emph{first}
dichotomy result for finite-valued VCSPs over non-Boolean domains.Comment: 15 page
The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case
The constraint satisfaction probem (CSP) is a well-acknowledged framework in
which many combinatorial search problems can be naturally formulated. The CSP
may be viewed as the problem of deciding the truth of a logical sentence
consisting of a conjunction of constraints, in front of which all variables are
existentially quantified. The quantified constraint satisfaction problem (QCSP)
is the generalization of the CSP where universal quantification is permitted in
addition to existential quantification. The general intractability of these
problems has motivated research studying the complexity of these problems under
a restricted constraint language, which is a set of relations that can be used
to express constraints.
This paper introduces collapsibility, a technique for deriving positive
complexity results on the QCSP. In particular, this technique allows one to
show that, for a particular constraint language, the QCSP reduces to the CSP.
We show that collapsibility applies to three known tractable cases of the QCSP
that were originally studied using disparate proof techniques in different
decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan 1979), Quantified
Horn-SAT (Karpinski, Kleine B\"{u}ning, and Schmitt 1987), and Quantified
Affine-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals
common structure among these cases, which are describable by constraint
languages over a two-element domain. In addition to unifying these known
tractable cases, we study constraint languages over domains of larger size
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