Fixed-Parameter Tractable (FPT)

Top PDF Fixed-Parameter Tractable (FPT):

Hitting topological minor models in planar graphs is fixed parameter tractable

Hitting topological minor models in planar graphs is fixed parameter tractable

deletion of k vertices. In other words, the class P represents some desired property that we want to impose to the input graph after deleting k vertices. This is a general graph modification problem with great expressive power as it encompasses many problems, depending on the choice of the property P. Unfortunately for most instantiations of P, this problem is not expected to admit a polynomial time algorithm. Lewis and Yannakakis showed in [ 21 ] that for any non- trivial and hereditary graph class P, the P-vertex deletion problem is NP-complete. Given this hardness result, an attractive alternative is to consider the standard parameterized version of the problem, called p -P-deletion where the parameter is the number k of vertex deletions. In this case the challenge is to investigate for which instantiations of P, p -P-deletion is fixed parameter tractable (or, in short, is FPT), i.e., it can be solved by an O k (n c )-time algorithm 1 (or FPT-algorithm), for some constant c. There is a long line of research on this general question. In many case, this concernins particular properties and possible optimizations of the contribution of k in the function hidden in the “O k ” notation (see e.g. [ 3 ]). However, it is interesting to notice that FPT-algorithms exist for general families of properties. In this direction the more general (and compact) results concern properties P that can be characterized by the exclusion of some finite set F of graphs (i.e., of size bounded by some constant h) with respect to some partial ordering relation ≤. We define
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Fixed-parameter tractable sampling for RNA design with multiple target structures

Fixed-parameter tractable sampling for RNA design with multiple target structures

for typical instances. For any fixed value of the treewidth, the complexity scales only linearly with the size of designed sequences and the number of targeted structures, i.e. our algorithms are fixed-parameter tractable (FPT). Remarkably, we could show that it is not possible to find a better, efficient method for sampling (unless P = NP), since the underlying counting problem is #P-hard. The practical relevance of this theoretical result is that it rules out substantially better sampling techniques. Even when using improved sampling methods, there will always remain an upper limit on the (in practice) tractable number and hetero- geneity of structures, the complexity of the directly treatable energy model, and the number and complexity of additional constraints that could be considered in future sampling-based applications. Technically, this result relies on a surprising bijection between valid sequences and independent sets of a bipartite graph, the latter being the object of recent breakthroughs in approximate counting complexity [ 20 , 21 ].
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A linear fixed parameter tractable algorithm for connected pathwidth

A linear fixed parameter tractable algorithm for connected pathwidth

LIRMM, Université de Montpellier, CNRS, France sedthilk@thilikos.info Abstract The graph parameter of pathwidth can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of node search where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one. Two desired characteristics for a cleaning strategy is to be monotone (no recontamination occurs) and connected (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called connected pathwidth. We prove that connected pathwidth is fixed parameter tractable, in particular we design a 2 O(k 2 ) · n time algorithm that checks whether the connected pathwidth of G is at most k. This resolves an open question by [Dereniowski, Osula, and Rzążewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019 ]. For our algorithm, we enrich the typical sequence technique that is able to deal with the connectivity demand. Typical sequences have been introduced in [Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996 ] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a “global” demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a 2 O(k 2 ) · n time algorithm for the monotone and connected version of the edge search number.
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Constructing Minimal Phylogenetic Networks from Softwired Clusters is Fixed Parameter Tractable

Constructing Minimal Phylogenetic Networks from Softwired Clusters is Fixed Parameter Tractable

recovered if the parameter is no longer allowed to appear in the exponent of |X | as it was in [20]. 1.2 Beyond softwired clusters: the wider context We believe that this approach is significant beyond the softwired cluster liter- ature. Other articles discuss the problem of constructing rooted phylogenetic networks not by combining clusters but by combining triplets [28, 30], charac- ters [12, 13, 35, 21] or entire phylogenetic trees into a network. These models are in general mutually distinct although they do have a significant common overlap which reaches its peak in the case of data derived from two phyloge- netic trees. To see this, note that if one takes the union of clusters represented by a set of two or more phylogenetic trees, then the reticulation number (or level) required to represent these clusters is in general less than or equal to the reticulation number (or level) required to topologically embed the trees themselves in the network, and this inequality is often strict. However, in the case of a set comprising exactly two trees the inequality becomes equality [29]. Hence for data obtained from two trees one could solve the reticulation number minimization and level minimization problems for clusters by using algorithms developed for the problem of topologically embedding the trees themselves into a network. These algorithms are highly efficient and fixed parameter tractable in a practical, as opposed to solely theoretical sense [1, 2, 5, 33]. However, these tree algorithms do not help us with more general cluster sets, because for more than two trees the optima of the cluster and tree models start to di- verge. Indeed, the cluster model often saves reticulations with respect to the tree model by weakening the concept of “above” and “below” in the network, which is exactly why the input tree topologies do not generally survive if one atomizes them into their constituent clusters [29]. Moreover, the literature on embedding three or more trees into a network is not yet mature, with articles restricting themselves to preliminary explorations [34, 17, 4]. It therefore seems plausible that the generator approach might be adapted to the tree model (or the other constructive methods mentioned) to yield a unified technique for producing positive complexity results for reticulation number minimization and level minimization, even in the case of many input trees (or data obtained from many input trees).
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Fixed-Parameter Tractable Sampling for RNA Design with Multiple Target Structures

Fixed-Parameter Tractable Sampling for RNA Design with Multiple Target Structures

Results: We revisit a central ingredient of most in-silico design methods: the sampling of sequences for the design of multi-target structures, possibly including pseudoknots. For this task, we present the efficient, tree decomposition-based algorithm RNARedPrint. Our fixed parameter tractable ap- proach is underpinned by establishing the #P-hardness of uniform sampling. Modeling the problem as a constraint network, RNARedPrint supports generic Boltzmann-weighted sampling for arbitrary additive RNA energy models; this enables the generation of RNA sequences meeting specific goals like expected free energies or GC-content. Finally, we empirically study general properties of the approach and generate biologically relevant multi-target Boltzmann-weighted designs for a com- mon design benchmark. Generating seed sequences with RNARedPrint, we demonstrate significant improvements over the previously best multi-target sampling strategy (uniform sampling). Availability: Our software is freely available at: https://github.com/yannponty/RNARedPrint Contact: yann.ponty@lix.polytechnique.fr
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Constant Thresholds Can Make Target Set Selection Tractable

Constant Thresholds Can Make Target Set Selection Tractable

Parameterized Complexity. One dimension of a parameterized problem is the input size s, and the other one is the parameter (usually a positive integer). A parameterized problem is called fixed-parameter tractable (fpt) with respect to a parameter k if it can be solved in f (k) · s O(1) time, where f is a computable function only depending on k. A core tool in the development of fixed-parameter algorithms is polynomial-time preprocessing by data reduction [ 3 , 13 ]. Here, the goal is to transform a given problem instance I with parameter k in polynomial time into an equivalent instance I 0 with parameter k 0 ≤ k such that the size of I 0 is upper-bounded by some function g only depending on k. If this is the case, we call I 0 a (problem) kernel of size g(k). Usually, this is achieved by applying polynomial-time executable data reduction rules. We call a data reduction rule R correct if the new instance I 0 that results from applying R to I is a yes-instance if and only if I is a yes-instance. An instance is called reduced with respect to some data reduction rule if further application of this rule has no effect on the instance. The whole process is called kernelization. 7
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On the fixed parameter tractability of agreement-based phylogenetic distances

On the fixed parameter tractability of agreement-based phylogenetic distances

Abstract Three important and related measures for summarizing the dissim- ilarity in phylogenetic trees are the minimum number of hybridization events required to fit two phylogenetic trees onto a single phylogenetic network (the hybridization number), the (rooted) subtree prune and regraft distance (the rSPR distance) and the tree bisection and reconnection distance (the TBR dis- tance) between two phylogenetic trees. The respective problems of computing these measures are known to be NP-hard, but also fixed-parameter tractable in their respective natural parameters. This means that, while they are hard to compute in general, for cases in which a parameter (here the hybridiza- tion number and rSPR/TBR distance, respectively) is small, the problem can be solved efficiently even for large input trees. Here, we present new analy- ses showing that the use of the “cluster reduction” rule – already defined for the hybridization number and the rSPR distance and introduced here for the TBR distance – can transform any O(f(p) · n)-time algorithm for any of these problems into an O(f(k) · n)-time one, where n is the number of leaves of the phylogenetic trees, p is the natural parameter and k is a much stronger (that
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Fixed-Parameter Algorithms For Protein Similarity Search Under mRNA Structure Constraints

Fixed-Parameter Algorithms For Protein Similarity Search Under mRNA Structure Constraints

Corollary 2. MRSO is solvable in O(2 12δ n) time. 3.1 Implied structure graphs with page-number two In light of algorithm A NEB and Lemma 1, a natural question to ask is whether MRSO is polynomial- time solvable in case we are provided an edge bipartition in which both parts have no edge crossings under the same vertex ordering. Such would be the case if G Γ had page-number two. In general, the page-number of a given graph G is the partitioning of E(G) into the smallest number of subsets possible, such that each subset of edges in the partition has no edge crossings under the same vertex ordering. If we could solve MRSO for page-number two graphs, we might also hope that MRSO becomes fixed-parameter tractable when parameterized by the page-number of G Γ . Unfortunately,
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On Backdoors To Tractable Constraint Languages

On Backdoors To Tractable Constraint Languages

to study the computational complexity of this problem, we usually consider backdoors with respect to a given tractable class T , i.e., such that all residual problems fall into the class T . In Boolean Satisfiability (SAT), it was shown that finding a minimum backdoor with respect to HornSAT, 2-SAT and their disjunction is fixed-parameter tractable with respect to the backdoor size [2][3]. It is significantly harder, however, to do so with respect to bounded treewidth formulas [4]. In this paper we study the computational complexity of finding a strong backdoor to a semantic tractable class of CSP, in the same spirit as a very recent work by Gaspers et al. [3]. Assuming that P 6= NP, semantic tractable classes are characterized by unions and intersections of languages of constraints closed by some operations. We make the following three main contributions:
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Fixed-point-like theorems on subspaces

Fixed-point-like theorems on subspaces

[12] Hirsch, M., Magill, M., Mas-Colell, A., A geometric approach to a class of equilibrium existence theorems, Journal of Mathematical Economics 19, (1990) 95-106. [13] Husseini, S. Y., Lasry, J.M., Magill, M., Existence of equilibrium with in- complete markets, Journal of Mathematical Economics 19, (1990) 39-67. [14] Kakutani, S., A generalization of Brouwer’s Fixed Point Theorem, Duke

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Finding Repeats With Fixed Gap

Finding Repeats With Fixed Gap

Unité de recherche INRIA Lorraine LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex France Unité de recherche[r]

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Fixed-point-like theorems on subspaces

Fixed-point-like theorems on subspaces

1. Introduction In this paper, we prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Let k be an in- teger and let V be a Euclidean space such that 0 ≤ k ≤ dimV, then the k-Grassmannian manifold of V, denoted G k (V), is the set of all the k-dimensional subspaces of V. The set G k (V) is a smooth compact manifold but, in general, it does not satisfy properties such as convexity or acyclicity and its Euler characteristic may be null. This prevents the use of classical fixed-point theorems as Brouwer’s [ 2 ], Kakutani’s [ 14 ], or Eilenberg- Montgomery’s theorem [ 7 ].
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Cosmological Parameter Estimation: Method

Cosmological Parameter Estimation: Method

In this way, we may write Eq. 3 in terms of the band–power and treat the latter as a parameter to be estimated. This then becomes the band–power likelihood function, L(δT fb ). Figure 1 shows the latest band power estimates of the CMB fluctuations. Some of the points have been obtained by maximizing this likelihood function; the errors are typically found by in a Bayesian approach, by integration in C b over L with a uniform prior (eg. DASI[12], VSA[13], CBI[14], ACBAR[15]). Other band powers and errors are estimated by using Monte Carlo based methods (see [26, 27]) like the WMAP[7], BOOMERANG[10], MAXIMA[11] and Archeops[9] ones. Notice that the variance due to the finite sample size (i.e., the sample variance, including the cosmic variance due to our observation of one realization of the sky) is fully incorporated into the analyses.
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On Homogeneous Distributed Parameter Systems

On Homogeneous Distributed Parameter Systems

where H and V are scalar functions of time and space variables. The quantity H(t, x) is the water level at the instant of time t ∈ R + in the point x ∈ R, and V (t, x) is the water velocity in the same position. The parameter g denotes the gravitation constant. Let us consider the case when the water channel is supported by two overflow spillways (Figure 1), which adjust an input and output flows in a pool (between spillways). The space argument is restricted on the segment [0, 1], where x = 0 and x = 1 are positions of spillways, and the equation (5) is supported with the boundary conditions [37]:
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Automated MOSFET parameter extraction

Automated MOSFET parameter extraction

After the test files are inserted and read by the newanal program, the program will first calculate the threshold voltage using maximum slope method (described in section 5.1), the [r]

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On fixed-point hardware polynomials

On fixed-point hardware polynomials

The reason for normalizing functions to unit intervals is that it greatly simplifies the arithmetic analysis needed to build an efficient architecture. This is indeed at the core of the contributions of this article. In other words, if a function has, in its original context, a different input range (for instance [0, 2 −k ] after range reduction in [2] or [4]), the present work claims that rewriting it as a function on [0, 1] or [−1, 1] is relevant for a fixed-point implementation.

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Parametrizations, fixed and random effects

Parametrizations, fixed and random effects

E of F relative to S and has the PDF proportional to exp{−q(s)/2σ 2 s }, where σ 2 s is some unknown positive parameter. We introduce the notion of bases separating the fixed and random effects and define comparison criteria between two separating bases using the partition functions and the maximum likelihood method. We illustrate our results for climate change detection using the set S of cubic splines. We show the influence of the choice of separating basis on the estimation of the linear tendency of the temperature

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Predicting the Best System Parameter Configuration: the (Per  Parameter Learning) PPL method

Predicting the Best System Parameter Configuration: the (Per Parameter Learning) PPL method

After indexing, the next step is to generate the set of K system configurations including the various values of the considered parameters in P. If all the possible configurations were generated, the chance of getting the best prediction would be increased. However, in practice, it is time consuming to do so (if not impossible for continuous parameters that can take their values in the range of integer or even real numbers). To solve this issue we generate what we think is a relatively high number of configurations by considering various values for each parameter and combining them (see Table 2 ). They are the predefined systems.
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Tractable sufficient stability conditions for a system coupling linear transport and differential equations

Tractable sufficient stability conditions for a system coupling linear transport and differential equations

m i ). This explains the term in e −2δxΛ −1 in functional (3). This functional is inspired by the complete Lyapunov-Krasovskii func- tional from [16] which is a necessary and sufficient condition for stability of linear systems with constant delay. It is however important to mention that the necessary conditions of [16] only states that if a linear time-delay sys- tems is asymptotically stable, there exists parameters P, Q, T, S and R such that the associated functional is a Lyapunov-Krasovskii functional for the systems. That being said, the method developed in [16] may not be used to derive a tractable stability test for a given system.
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Some tractable instances of interval data minmax regret problems: bounded distance from triviality

Some tractable instances of interval data minmax regret problems: bounded distance from triviality

[7] H. L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. on Computing, 25(6):1305–1317, 1996. [8] R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer, 1999. [9] B. Escoffier, J. Monnot, and O. Spanjaard. Some tractable instances of interval data minmax regret problems: bounded distance from trivial- ity. Technical Report 265, Cahiers de recherche, LAMSADE, 2007. URL http://www.lamsade.dauphine.fr/cahiers/PDF/cahierLamsade265.pdf. [10] J. Guo, F. H¨ uffner, and R. Niedermeier. A structural view on parameterizing

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