566 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
The complexity of Boolean surjective general-valued CSPs
Valued constraint satisfaction problems (VCSPs) are discrete optimisation
problems with a -valued objective function given as
a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on
labels from and an optimal assignment is required to use both
labels from . Examples include the classical global Min-Cut problem in
graphs and the Minimum Distance problem studied in coding theory.
We establish a dichotomy theorem and thus give a complete complexity
classification of Boolean surjective VCSPs with respect to exact solvability.
Our work generalises the dichotomy for -valued constraint
languages (corresponding to surjective decision CSPs) obtained by Creignou and
H\'ebrard. For the maximisation problem of -valued
surjective VCSPs, we also establish a dichotomy theorem with respect to
approximability.
Unlike in the case of Boolean surjective (decision) CSPs, there appears a
novel tractable class of languages that is trivial in the non-surjective
setting. This newly discovered tractable class has an interesting mathematical
structure related to downsets and upsets. Our main contribution is identifying
this class and proving that it lies on the borderline of tractability. A
crucial part of our proof is a polynomial-time algorithm for enumerating all
near-optimal solutions to a generalised Min-Cut problem, which might be of
independent interest.Comment: v5: small corrections and improved presentatio
Galois correspondence for counting quantifiers
We introduce a new type of closure operator on the set of relations,
max-implementation, and its weaker analog max-quantification. Then we show that
approximation preserving reductions between counting constraint satisfaction
problems (#CSPs) are preserved by these two types of closure operators.
Together with some previous results this means that the approximation
complexity of counting CSPs is determined by partial clones of relations that
additionally closed under these new types of closure operators. Galois
correspondence of various kind have proved to be quite helpful in the study of
the complexity of the CSP. While we were unable to identify a Galois
correspondence for partial clones closed under max-implementation and
max-quantification, we obtain such results for slightly different type of
closure operators, k-existential quantification. This type of quantifiers are
known as counting quantifiers in model theory, and often used to enhance first
order logic languages. We characterize partial clones of relations closed under
k-existential quantification as sets of relations invariant under a set of
partial functions that satisfy the condition of k-subset surjectivity. Finally,
we give a description of Boolean max-co-clones, that is, sets of relations on
{0,1} closed under max-implementations.Comment: 28 pages, 2 figure
The Complexity of Surjective Homomorphism Problems -- a Survey
We survey known results about the complexity of surjective homomorphism
problems, studied in the context of related problems in the literature such as
list homomorphism, retraction and compaction. In comparison with these
problems, surjective homomorphism problems seem to be harder to classify and we
examine especially three concrete problems that have arisen from the
literature, two of which remain of open complexity
Comment on "Control landscapes are almost always trap free: a geometric assessment"
We analyze a recent claim that almost all closed, finite dimensional quantum
systems have trap-free (i.e., free from local optima) landscapes (B. Russell
et.al. J. Phys. A: Math. Theor. 50, 205302 (2017)). We point out several errors
in the proof which compromise the authors' conclusion.
Interested readers are highly encouraged to take a look at the "rebuttal"
(see Ref. [1]) of this comment published by the authors of the criticized work.
This "rebuttal" is a showcase of the way the erroneous and misleading
statements under discussion will be wrapped up and injected in their future
works, such as R. L. Kosut et.al, arXiv:1810.04362 [quant-ph] (2018).Comment: 6 pages, 1 figur
On the reduction of the CSP dichotomy conjecture to digraphs
It is well known that the constraint satisfaction problem over general
relational structures can be reduced in polynomial time to digraphs. We present
a simple variant of such a reduction and use it to show that the algebraic
dichotomy conjecture is equivalent to its restriction to digraphs and that the
polynomial reduction can be made in logspace. We also show that our reduction
preserves the bounded width property, i.e., solvability by local consistency
methods. We discuss further algorithmic properties that are preserved and
related open problems.Comment: 34 pages. Article is to appear in CP2013. This version includes two
appendices with proofs of claims omitted from the main articl
On the Approximation of Constrained Linear Quadratic Regulator Problems and their Application to Model Predictive Control - Supplementary Notes
By parametrizing input and state trajectories with basis functions different
approximations to the constrained linear quadratic regulator problem are
obtained. These notes present and discuss technical results that are intended
to supplement a corresponding journal article. The results can be applied in a
model predictive control context.Comment: 19 pages, 1 figur
Robustly Solvable Constraint Satisfaction Problems
An algorithm for a constraint satisfaction problem is called robust if it
outputs an assignment satisfying at least -fraction of the
constraints given a -satisfiable instance, where
as . Guruswami and
Zhou conjectured a characterization of constraint languages for which the
corresponding constraint satisfaction problem admits an efficient robust
algorithm. This paper confirms their conjecture
Deterministic counting of graph colourings using sequences of subgraphs
In this paper we propose a deterministic algorithm for approximately counting
the -colourings of sparse random graphs . In particular, our
algorithm computes in polynomial time a approximation of
the logarithm of the number of -colourings of for with high probability over the graph instances.
Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA
'06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation
decay} to compute {\em deterministically} marginals of {\em Gibbs
distribution}. We develop a scheme whose accuracy depends on {\em
non-reconstruction} of the colourings of , rather than {\em
uniqueness} that are required in previous works. This leaves open the
possibility for our schema to be sufficiently accurate even for .
The set up for establishing correlation decay is as follows: Given
, we alter the graph structure in some specific region of
the graph by deleting edges between vertices of . Then we show that
the effect of this change on the marginals of Gibbs distribution, diminishes as
we move away from . Our approach is novel and suggests a new context
for the study of deterministic counting algorithms
- …