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R. Ganian:
"Discovering Archipelagos of Tractability for Constraint Satisfaction and Counting";
Talk: Contemporary Trends in Theoretical Computer Science 2015, Prag, Tschechien (invited); 2015-06-26 - 2015-06-27.

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 paper 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 isn't 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,
i.e., 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 FPT time a strong backdoor into
a scattered class (associated with a list of finite conservative
constraint languages) of size k or correctly decides that there
isn't 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.