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R. Ganian, M. S. Ramanujan, S. Szeider:
"Discovering Archipelagos of Tractability for Constraint Satisfaction and Counting";
ACM Transactions on Algorithms, 13 (2017), S. 1 - 31.

The Constraint Satisfaction Problem (CSP) is a central and generic
computational problem which provides a common framework for many
theoretical and practical applications. A central line of research is
concerned with the identification of classes of instances for which
CSP can be solved in polynomial time; such classes are often called
"islands of tractability." A prominent way of defining islands of
tractability for CSP is to restrict the relations that may occur in
the constraints to a fixed set, called a constraint language, whereas
a constraint language is conservative if it contains all unary
relations. Schaefer´s famous Dichotomy Theorem (STOC 1978) identifies
all islands of tractability in terms of tractable constraint languages
over a Boolean domain of values. Since then, many extensions and
generalizations of this result have been obtained. Recently, Bulatov
(TOCL 2011, JACM 2013) gave a full characterization of all islands of
tractability for CSP and the counting version #CSP that are defined in
terms of conservative constraint languages. This article addresses
the general limit of the mentioned tractability results for CSP and
#CSP, that they only apply to instances where all constraints belong
to a single tractable language (in general, the union of two tractable
languages is not tractable). We show that we can overcome this
limitation as long as we keep some control of how constraints over the
various considered tractable languages interact with each other. For
this purpose, we utilize the notion of a strong backdoor of a CSP
instance, as introduced by Williams et al. (IJCAI 2003), which is a
set of variables that when instantiated, moves the instance to an
island of tractability, that is, to a tractable class of instances. We
consider strong backdoors into scattered classes, consisting of CSP
instances where each connected component belongs entirely to some
class from a list of tractable classes. Figuratively speaking, a
scattered class constitutes an archipelago of tractability. The main
difficulty lies in finding a strong backdoor of given size k; once it
is found, we can try all possible instantiations of the backdoor
variables and apply the polynomial time algorithms associated with the
islands of tractability on the list component-wise. Our main result is
an algorithm that, given a CSP instance with n variables, finds in
time f (k)nO(1) a strong backdoor into a scattered class (associated
with a list of finite conservative constraint languages) of size k or
correctly decides that there is not such a backdoor. This also gives
the running time for solving (#)CSP, provided that (#)CSP is
polynomial-time tractable for the considered constraint languages. Our
result makes significant progress towards the main goal of the
backdoor-based approach to CSPs-the identification of maximal base
classes for which small backdoors can be detected efficiently.

http://dx.doi.org/10.1145/3014587

https://dl.acm.org/citation.cfm?doid=3040971.3014587