14,508 research outputs found
Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data
Constraint Programming (CP) has proved an effective paradigm to model and
solve difficult combinatorial satisfaction and optimisation problems from
disparate domains. Many such problems arising from the commercial world are
permeated by data uncertainty. Existing CP approaches that accommodate
uncertainty are less suited to uncertainty arising due to incomplete and
erroneous data, because they do not build reliable models and solutions
guaranteed to address the user's genuine problem as she perceives it. Other
fields such as reliable computation offer combinations of models and associated
methods to handle these types of uncertain data, but lack an expressive
framework characterising the resolution methodology independently of the model.
We present a unifying framework that extends the CP formalism in both model
and solutions, to tackle ill-defined combinatorial problems with incomplete or
erroneous data. The certainty closure framework brings together modelling and
solving methodologies from different fields into the CP paradigm to provide
reliable and efficient approches for uncertain constraint problems. We
demonstrate the applicability of the framework on a case study in network
diagnosis. We define resolution forms that give generic templates, and their
associated operational semantics, to derive practical solution methods for
reliable solutions.Comment: Revised versio
Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP
Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's
Unique Games Conjecture (STOC 2002) is true, then for every constraint
satisfaction problem (CSP), the best approximation ratio is attained by a
certain simple semidefinite programming and a rounding scheme for it. In this
paper, we show that similar results hold for constant-time approximation
algorithms in the bounded-degree model. Specifically, we present the
followings: (i) For every CSP, we construct an oracle that serves an access, in
constant time, to a nearly optimal solution to a basic LP relaxation of the
CSP. (ii) Using the oracle, we give a constant-time rounding scheme that
achieves an approximation ratio coincident with the integrality gap of the
basic LP. (iii) Finally, we give a generic conversion from integrality gaps of
basic LPs to hardness results. All of those results are \textit{unconditional}.
Therefore, for every bounded-degree CSP, we give the best constant-time
approximation algorithm among all. A CSP instance is called -far from
satisfiability if we must remove at least an -fraction of constraints
to make it satisfiable. A CSP is called testable if there is a constant-time
algorithm that distinguishes satisfiable instances from -far
instances with probability at least . Using the results above, we also
derive, under a technical assumption, an equivalent condition under which a CSP
is testable in the bounded-degree model
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