Kernelization – Preprocessing with a Guarantee

The F-Minor-Free Deletion problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. It generalizes classic graph problems such as Vertex Cover and Feedback Vertex Set. This paper analyzes to what extent provably effective and efficient preprocessing is possible for F-Minor-Free

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“…For further background on parameterized complexity and kernelization we refer to one of the textbooks [14,16,32] or recent surveys [5,27,30].…”

Section: Parameterized Complexity and Kernelizationmentioning

confidence: 99%

Uniform Kernelization Complexity of Hitting Forbidden Minors

Giannopoulou

Jansen

Lokshtanov

et al. 2017

ACM Trans. Algorithms

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“…Conversely, by giving PPTs from problems that are already known not to admit polynomial compressions (under some assumption) we rule out polynomial kernels for the target problems. For more background on kernelization we refer the reader to the recent survey by Lokshtanov et al [15].…”

Section: Parameterized Complexity and Kernelizationmentioning

confidence: 99%

Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem

2015

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The workflow satisfiability problem (wsp) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan -an assignment of tasks to authorized userssuch that all constraints are satisfied.The wsp is, in fact, the conservative constraint satisfaction problem (i.e., for each variable, here called task, we have a unary authorization constraint) and is, thus, NP-complete. It was observed by Wang and Li (2010) that the number k of tasks is often quite small and so can be used as a parameter, and several subsequent works have studied the parameterized complexity of wsp regarding parameter k.We take a more detailed look at the kernelization complexity of wsp(Γ) when Γ denotes a finite or infinite set of allowed constraints. Our main result is a dichotomy for the case that all constraints in Γ are regular: (1) We are able to reduce the number n of users to n ′ ≤ k. This entails a kernelization to size poly(k) for finite Γ, and, under mild technical conditions, to size poly(k + m) for infinite Γ, where m denotes the number of constraints. (2) Already wsp(R) for some R ∈ Γ allows no polynomial kernelization in k + m unless the polynomial hierarchy collapses.

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“…If this is the case, then one says that L is fixedparameter tractable for the parameter k. The corresponding complexity class is called FPT. If L could only be solved in polynomial running time where the degree of the polynomial depends on k (such as x f (k) ), then, for parameter k, the problem L is said to lie in the-strictly larger [13] A core tool in the development of fixed-parameter algorithms is polynomialtime preprocessing by data reduction [8,17,21]. Here, the goal is to transform a given problem instance I with parameter k in polynomial time into an equivalent instance I with parameter k ≤ k such that the size of I is upper-bounded by some function g only depending on k. If this is the case, then we call I a (problem) kernel of size g(k).…”

Section: Inputmentioning

confidence: 99%

Using Patterns to Form Homogeneous Teams

et al. 2013

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Homogeneous team formation is the task of grouping individuals into teams, each of which consists of members who fulfill the same set of prespecified properties. In this theoretical work, we propose, motivate, and analyze a combinatorial model where, given a matrix over a finite alphabet whose rows correspond to individuals and columns correspond to attributes of individuals, the user specifies lower and upper bounds on team sizes as well as combinations of attributes that have to be homogeneous (that is, identical) for all members of the corresponding teams. Furthermore, the user can define a cost for assigning any individual to a certain team. We show that some special cases of our new model lead to NP-hard problems while others allow for (fixed-parameter) tractability results. For example, the problem is already That version concentrates on the anonymization aspects of the model. In our new version we slightly extend our model and show how it applies to (homogeneous) clustering of individuals, that is, to homogeneous team formation. Indeed, we now claim that the models and ideas better fit with these applications than with the previous data anonymization motivation. Apart from full proofs omitted in the extended abstract and also adapting our old ideas to the new extended model, the current article also contains a new and easier proof of NP-hardness, a new proof for showing that polynomial-time data reduction in term of so-called polynomial-size problem kernels is unlikely to exist with respect to certain parameterizations, and a new algorithm for the (still NP-hard) special case ignoring costs. Many of the new findings are part of the diploma thesis [18] NP-hard even if (i) there are no lower and upper bounds on the team sizes, (ii) all costs are zero, and (iii) the matrix has only two columns. In contrast, the problem becomes fixed-parameter tractable for the combined parameter "number of possible teams" and "number of different individuals", the latter being upper-bounded by the number of rows.

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Kernelization – Preprocessing with a Guarantee

Cited by 52 publications

References 58 publications

Uniform Kernelization Complexity of Hitting Forbidden Minors

Uniform Kernelization Complexity of Hitting Forbidden Minors

Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem

Using Patterns to Form Homogeneous Teams

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