An Overview of Techniques for Designing Parameterized Algorithms

Sloper, Christian; Telle, Jan Arne

doi:10.1093/comjnl/bxm038

Cited by 43 publications

(18 citation statements)

References 34 publications

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“…is an arbitrary computable function and α is a constant. If no such algorithm exists then P is said to be fixed-parameter (fp-)intractable for parameter set K .Proving fixed-parameter tractability is conceptually straightforward: It suffices to produce just one algorithm that computes the problem in fixed-parameter tractable time (see, e.g., Sloper and Telle, 2008, for a review of generic techniques for building such algorithms).W[1]-hard problems are problems, including a set of parameters, that cannot 2 be solved in fixed parameter time even when all parameters in their set are small, and hence are fixed-parameter intractable according to above definition. A problem K 1 - P 1 can be proven W[1]-hard, by taking a known W[1]-hard problem K 2 - P 2 and parameterized reducing it to the problem K 1 - P 1 .…”

Section: Preliminaries From Bayesian Modelingmentioning

confidence: 99%

Intentional Communication: Computationally Easy or Difficult?

Rooij¹,

Kwisthout²,

Blokpoel³

et al. 2011

Front. Hum. Neurosci.

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Human intentional communication is marked by its flexibility and context sensitivity. Hypothesized brain mechanisms can provide convincing and complete explanations of the human capacity for intentional communication only insofar as they can match the computational power required for displaying that capacity. It is thus of importance for cognitive neuroscience to know how computationally complex intentional communication actually is. Though the subject of considerable debate, the computational complexity of communication remains so far unknown. In this paper we defend the position that the computational complexity of communication is not a constant, as some views of communication seem to hold, but rather a function of situational factors. We present a methodology for studying and characterizing the computational complexity of communication under different situational constraints. We illustrate our methodology for a model of the problems solved by receivers and senders during a communicative exchange. This approach opens the way to a principled identification of putative model parameters that control cognitive processes supporting intentional communication.

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Section: Preliminaries From Bayesian Modelingmentioning

confidence: 99%

Intentional Communication: Computationally Easy or Difficult?

Rooij¹,

Kwisthout²,

Blokpoel³

et al. 2011

Front. Hum. Neurosci.

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“…It should be noted that both the kernel and the algorithm for p-Set Splitting presented here also work for the p-Not All Equal SAT problem. The reduction rule we use to handle instances with strong cut-sets has similarities with reduction rules based on crown decompositions [3,8,19], and it seems that crown decompositions and strong cut-sets are closely related. This similarity also makes us believe that the duality theorem we made us of in our kenrelization algorithm will be a useful tool in the filed of kernelization.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…The correctness proof for this reduction rule is simple, and given for example in [19] for the case of p-Max Cut.…”

Section: Lemmamentioning

confidence: 99%

Even Faster Algorithm for Set Splitting!

Lokshtanov

Saurabh

2009

Parameterized and Exact Computation

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In the p-Set Splitting problem we are given a universe U , a family F of subsets of U and a positive integer k and the objective is to find a partition of U into W and B such that there are at least k sets in F that have non-empty intersection with both B and W . In this paper we study p-Set Splitting from the view point of kernelization and parameterized algorithms. Given an instance (U, F, k) of p-Set Splitting, our kernelization algorithm obtains an equivalent instance with at most 2k sets and k elements in polynomial time. Finally, we give a fixed parameter tractable algorithm for p-Set Splitting running in time O(1.9630 k + N ), where N is the size of the instance. Both our kernel and our algorithm improve over the best previously known results. Our kernelization algorithm utilizes a classical duality theorem for a connectivity notion in hypergraphs. We believe that the duality theorem we make use of could become an important tool in obtaining kernelization algorithms.

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“…The first problem we consider is the natural parameterization of the Sized Crown problem previously proven to be NP-complete by Sloper [36]. This decision problem involves determining if a graph contains a crown of a certain size.…”

Section: Relation To Crownsmentioning

confidence: 99%

“…The parameterized complexity of these problems is mentioned as an open problem by respectively Sloper [36] and Abu-Khzam et al [35] and to our knowledge these problems have not been solved before.…”

Section: Minimum Crownmentioning

confidence: 99%

Bipartite graphs and the decomposition of systems of equations

Bomhoff

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Solving large systems of equations is a problem often encountered in engineering disciplines. However, as such systems grow, the effort required for finding a solution to them increases as well. In order to be able to cope with ever larger systems of equations, some form of decomposition is needed. By decomposing a large system into smaller subsystems, the total effort required for finding a solution may decrease. However, whether this is really the case of course depends on the additional effort required for obtaining the decomposition itself. In this thesis several aspects of the difficulty of obtaining such decompositions are explored.The first problem discussed in this thesis is that of the decomposition of a system of non-linear equations. After describing the well-known conditions for consistency in systems of equations, the decomposition problem for under-specified systems is analyzed. This analysis, based on the existing notion of free square blocks, leads to several W [1]-hardness results. The most important of which is the proof that finding a decomposition into subsystems where the largest subsystem is as small as possible is W [1]-hard. This implies that even if the size of such a largest subsystem is bounded by a constant, the problem is still not solvable in polynomial time under the current working hypothesis that W [1] FP T . As a by-product of these results two open problems regarding crown structures for the vertex cover problem are settled.Having investigated the non-linear case, attention is then shifted to the case of systems of linear equations. Such systems are commonly represented as matrices upon which pivot operations are performed. In such operations preserving sparsity of the input matrix is often beneficial to limit both storage and processing requirements. The choice of pivot elements can strongly influence the level of sparsity that is preserved. Changing a zero value in a matrix to a non-zero value is called fill-in. An important role in preserving sparsity is played by bisimplicial edges in the bipartite graph that corresponds naturally to a matrix. Such bisimplicial edges correspond to pivots in the matrix that completely avoid fill-in. In this thesis a new O (nm) algorithm for finding such bisimplicial edges for an n×n matrix with m non-zero values is described and analyzed on a common class vii of random matrices. It is shown that the expected running-time of the algorithm on matrices corresponding to bipartite graphs from the G n,n,p model is O n 2 , which is linear in the input size.After analyzing single pivots that avoid turning zero elements into nonzero elements, the class of matrices that allow Gaussian elimination using only such pivots is discussed. Such matrices correspond to prefect elimination bipartite graphs. Several algorithms are known for the recognition of this class of graphs or matrices, but most of them are based on matrix multiplication implying sparse input matrices may still result in dense matrices along the way. In this thesis two new algorith...

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An Overview of Techniques for Designing Parameterized Algorithms

Cited by 43 publications

References 34 publications

Intentional Communication: Computationally Easy or Difficult?

Intentional Communication: Computationally Easy or Difficult?

Even Faster Algorithm for Set Splitting!

Bipartite graphs and the decomposition of systems of equations

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