Backdoors into heterogeneous classes of SAT and CSP

Gaspers, Serge; Misra, Neeldhara; Ordyniak, Sebastian; Szeider, Stefan; Živný, Stanislav

doi:10.1016/j.jcss.2016.10.007

Cited by 19 publications

(26 citation statements)

References 28 publications

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“…One advantage of parameterizing by the size of modulators is that it allows us to build on the vast array of research of polynomial-time algorithms on specific graph classes (see, for instance, [10,33]). In other fields of computer science, modulators are often called backdoors and have been successfully used to obtain efficient algorithms for, e.g., satisfiability and constraint satisfaction [21,22].…”

Section: A: Width Measuresmentioning

confidence: 99%

Solving Problems on Graphs of High Rank-Width

2017

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A modulator in a graph is a vertex set whose deletion places the considered graph into some specified graph class. The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain fixed-parameter algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are more powerful for fixed-parameter algorithms than the cardinality of modulators and rank-width itself. Then, we develop a fixed-parameter algorithm for finding such wellstructured modulators to every graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in monadic second order logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

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Section: A: Width Measuresmentioning

confidence: 99%

Solving Problems on Graphs of High Rank-Width

2017

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“…Extensions to the basic notion of a backdoor have been proposed, including backdoors with empty clause detection [6], backdoors in the context of learning [7], heterogeneous backdoors where different instantiations of the backdoor variables may result in instances that belong to different base classes [15], and backdoors into scattered classes where each connected component of an instance may belong to a different tractable class [14]. Le Bras et al [20] used backdoors to speed-up the solution of hard problems in materials discovery, using a crowd sourcing approach to find small backdoors.…”

Section: Related Workmentioning

confidence: 99%

Backdoors to Tractable Valued CSP

Ganian

Ramanujan

Szeider

2016

Lecture Notes in Computer Science

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Abstract. We extend the notion of a strong backdoor from the CSP setting to the Valued CSP setting (VCSP, for short). This provides a means for augmenting a class of tractable VCSP instances to instances that are outside the class but of small distance to the class, where the distance is measured in terms of the size of a smallest backdoor. We establish that VCSP is fixed-parameter tractable when parameterized by the size of a smallest backdoor into every tractable class of VCSP instances characterized by a (possibly infinite) tractable valued constraint language of finite arity and finite domain. We further extend this fixed-parameter tractability result to so-called "scattered classes" of VCSP instances where each connected component may belong to a different tractable class.

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“…The parameterized complexity of strong backdoor detection when the target class is only defined by restrictions on the constraint language is W[2]-hard for every tractable class defined by idempotent polymorphisms even if the parameter is d + k, where d is the domain size and k is the size of the strong backdoor [91]. Nonetheless, if r is the arity of the instance, the problem becomes FPT for the parameter d + k + r [91] and even k + r if the tractable class is defined by finitely many polymorphisms [37].…”

Section: Backdoorsmentioning

confidence: 99%

“…Nonetheless, if r is the arity of the instance, the problem becomes FPT for the parameter d + k + r [91] and even k + r if the tractable class is defined by finitely many polymorphisms [37]. However, the classes for which strong backdoor detection is FPT in k + r are quite small and for all tractable classes represented in Fig.…”

Section: Backdoorsmentioning

confidence: 99%

“…The construction is based on the observation that if we have a partition (C 1 , C 2 ) of the constraints such that C 1 ∈ T , then the vertex cover of the primal graph of C 2 is a strong backdoor to T ; the minimumsize partition backdoor is then defined as the minimum-size such backdoor over every possible partition of the constraints. This kind of backdoor is easier to compute (FPT in k + l, where k is the size of the partition backdoor and l is the size of the language of the input instance), but can be arbitrarily larger than the minimum-size strong backdoor [91].…”

Section: Backdoorsmentioning

confidence: 99%

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Tractability in constraint satisfaction problems: a survey

Carbonnel

Cooper

2015

Constraints

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Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. What is tractability?The idea that an algorithm is efficient if its time complexity is a polynomial function of the size of its input can be traced back to pioneering work of Cobham [45] and Edmonds [83], but the foundations of complexity theory are based on the seminal work of Cook [52] and Karp [118]. A computational decision problem (such as the constraint satisfaction problem or CSP) consists of a generic instance (in the case of the CSP, a set of variables, their

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Backdoors into heterogeneous classes of SAT and CSP

Cited by 19 publications

References 28 publications

Solving Problems on Graphs of High Rank-Width

Solving Problems on Graphs of High Rank-Width

Backdoors to Tractable Valued CSP

Tractability in constraint satisfaction problems: a survey

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