93,471 research outputs found
Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning under structural
restrictions. All these problems involve two tasks: (i) identifying the
structure in the input as required by the restriction, and (ii) using the
identified structure to solve the reasoning task efficiently. We show that for
most of the considered problems, task (i) admits a polynomial-time
preprocessing to a problem kernel whose size is polynomial in a structural
problem parameter of the input, in contrast to task (ii) which does not admit
such a reduction to a problem kernel of polynomial size, subject to a
complexity theoretic assumption. As a notable exception we show that the
consistency problem for the AtMost-NValue constraint admits a polynomial kernel
consisting of a quadratic number of variables and domain values. Our results
provide a firm worst-case guarantees and theoretical boundaries for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541,
arXiv:1104.556
The Role of Commutativity in Constraint Propagation Algorithms
Constraint propagation algorithms form an important part of most of the
constraint programming systems. We provide here a simple, yet very general
framework that allows us to explain several constraint propagation algorithms
in a systematic way. In this framework we proceed in two steps. First, we
introduce a generic iteration algorithm on partial orderings and prove its
correctness in an abstract setting. Then we instantiate this algorithm with
specific partial orderings and functions to obtain specific constraint
propagation algorithms.
In particular, using the notions commutativity and semi-commutativity, we
show that the {\tt AC-3}, {\tt PC-2}, {\tt DAC} and {\tt DPC} algorithms for
achieving (directional) arc consistency and (directional) path consistency are
instances of a single generic algorithm. The work reported here extends and
simplifies that of Apt \citeyear{Apt99b}.Comment: 35 pages. To appear in ACM TOPLA
Dynamic Consistency of Conditional Simple Temporal Networks via Mean Payoff Games: a Singly-Exponential Time DC-Checking
Conditional Simple Temporal Network (CSTN) is a constraint-based
graph-formalism for conditional temporal planning. It offers a more flexible
formalism than the equivalent CSTP model of Tsamardinos, Vidal and Pollack,
from which it was derived mainly as a sound formalization. Three notions of
consistency arise for CSTNs and CSTPs: weak, strong, and dynamic. Dynamic
consistency is the most interesting notion, but it is also the most challenging
and it was conjectured to be hard to assess. Tsamardinos, Vidal and Pollack
gave a doubly-exponential time algorithm for deciding whether a CSTN is
dynamically-consistent and to produce, in the positive case, a dynamic
execution strategy of exponential size. In the present work we offer a proof
that deciding whether a CSTN is dynamically-consistent is coNP-hard and provide
the first singly-exponential time algorithm for this problem, also producing a
dynamic execution strategy whenever the input CSTN is dynamically-consistent.
The algorithm is based on a novel connection with Mean Payoff Games, a family
of two-player combinatorial games on graphs well known for having applications
in model-checking and formal verification. The presentation of such connection
is mediated by the Hyper Temporal Network model, a tractable generalization of
Simple Temporal Networks whose consistency checking is equivalent to
determining Mean Payoff Games. In order to analyze the algorithm we introduce a
refined notion of dynamic-consistency, named \epsilon-dynamic-consistency, and
present a sharp lower bounding analysis on the critical value of the reaction
time \hat{\varepsilon} where the CSTN transits from being, to not being,
dynamically-consistent. The proof technique introduced in this analysis of
\hat{\varepsilon} is applicable more in general when dealing with linear
difference constraints which include strict inequalities
Checking Dynamic Consistency of Conditional Hyper Temporal Networks via Mean Payoff Games (Hardness and (pseudo) Singly-Exponential Time Algorithm)
In this work we introduce the \emph{Conditional Hyper Temporal Network
(CHyTN)} model, which is a natural extension and generalization of both the
\CSTN and the \HTN model. Our contribution goes as follows. We show that
deciding whether a given \CSTN or CHyTN is dynamically consistent is
\coNP-hard. Then, we offer a proof that deciding whether a given CHyTN is
dynamically consistent is \PSPACE-hard, provided that the input instances are
allowed to include both multi-head and multi-tail hyperarcs. In light of this,
we continue our study by focusing on CHyTNs that allow only multi-head or only
multi-tail hyperarcs, and we offer the first deterministic (pseudo)
singly-exponential time algorithm for the problem of checking the
dynamic-consistency of such CHyTNs, also producing a dynamic execution strategy
whenever the input CHyTN is dynamically consistent. Since \CSTN{s} are a
special case of CHyTNs, this provides as a byproduct the first
sound-and-complete (pseudo) singly-exponential time algorithm for checking
dynamic-consistency in CSTNs. The proposed algorithm is based on a novel
connection between CSTN{s}/CHyTN{s} and Mean Payoff Games. The presentation of
the connection between \CSTN{s}/CHyTNs and \MPG{s} is mediated by the \HTN
model. In order to analyze the algorithm, we introduce a refined notion of
dynamic-consistency, named -dynamic-consistency, and present a sharp
lower bounding analysis on the critical value of the reaction time
where a \CSTN/CHyTN transits from being, to not being,
dynamically consistent. The proof technique introduced in this analysis of
is applicable more generally when dealing with linear
difference constraints which include strict inequalities.Comment: arXiv admin note: text overlap with arXiv:1505.0082
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