fixed parameter tractable
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2021 ◽  
Author(s):  
Bertrand Marchand ◽  
Yann Ponty ◽  
Laurent Bulteau

Abstract Hard graph problems are ubiquitous in Bioinformatics, inspiring the design of specialized Fixed-Parameter Tractable algorithms, many of which rely on a combination of tree-decomposition and dynamic programming. The time/space complexities of such approaches hinge critically on low values for the treewidth tw of the input graph. In order to extend their scope of applicability, we introduce the Tree-Diet problem, i.e. the removal of a minimal set of edges such that a given tree-decomposition can be slimmed down to a prescribed treewidth tw. Our rationale is that the time gained thanks to a smaller treewidth in a parameterized algorithm compensates the extra post-processing needed to take deleted edges into account. Our core result is an FPT dynamic programming algorithm for Tree-Diet, using 2^O(tw)n time and space. We complement this result with parameterized complexity lower-bounds for stronger variants (e.g., NP-hardness when tw or tw − tw is constant). We propose a prototype implementation for our approach which we apply on difficult instances of selected RNA-based problems: RNA design, sequence-structure alignment, and search of pseudoknotted RNAs in genomes, revealing very encouraging results. This work paves the way for a wider adoption of tree-decomposition-based algorithms in Bioinformatics.


2021 ◽  
Vol 13 (3) ◽  
pp. 1-16
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
Daniel Lokshtanov ◽  
Fahad Panolan ◽  
Saket Saurabh ◽  
...  

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.


2021 ◽  
Author(s):  
Anneke Haga ◽  
Carsten Lutz ◽  
Leif Sabellek ◽  
Frank Wolter

We introduce and study several notions of approximation for ontology-mediated queries based on the description logics ALC and ALCI. Our approximations are of two kinds: we may (1) replace the ontology with one formulated in a tractable ontology language such as ELI or certain TGDs and (2) replace the database with one from a tractable class such as the class of databases whose treewidth is bounded by a constant. We determine the computational complexity and the relative completeness of the resulting approximations. (Almost) all of them reduce the data complexity from coNP-complete to PTime, in some cases even to fixed-parameter tractable and to linear time. While approximations of kind (1) also reduce the combined complexity, this tends to not be the case for approximations of kind (2). In some cases, the combined complexity even increases.


2021 ◽  
Vol 24 (3) ◽  
pp. 1-22
Author(s):  
Jason Crampton ◽  
Gregory Z. Gutin ◽  
Diptapriyo Majumdar

User authorization queries in the context of role-based access control have attracted considerable interest in the past 15 years. Such queries are used to determine whether it is possible to allocate a set of roles to a user that enables the user to complete a task, in the sense that all the permissions required to complete the task are assigned to the roles in that set. Answering such a query, in general, must take into account a number of factors, including, but not limited to, the roles to which the user is assigned and constraints on the sets of roles that can be activated. Answering such a query is known to be NP-hard. The presence of multiple parameters and the need to find efficient and exact solutions to the problem suggest that a multi-variate approach will enable us to better understand the complexity of the user authorization query problem (UAQ). In this article, we establish a number of complexity results for UAQ. Specifically, we show the problem remains hard even when quite restrictive conditions are imposed on the structure of the problem. Our fixed-parameter tractable (FPT) results show that we have to use either a parameter with potentially quite large values or quite a restricted version of UAQ. Moreover, our second FPT algorithm is complex and requires sophisticated, state-of-the-art techniques. In short, our results show that it is unlikely that all variants of UAQ that arise in practice can be solved reasonably quickly in general.


Author(s):  
Niels Grüttemeier ◽  
Christian Komusiewicz ◽  
Nils Morawietz

A Bayesian network is a directed acyclic graph that represents statistical dependencies between variables of a joint probability distribution. A fundamental task in data science is to learn a Bayesian network from observed data. Polytree Learning is the problem of learning an optimal Bayesian network that fulfills the additional property that its underlying undirected graph is a forest. In this work, we revisit the complexity of Polytree Learning. We show that Polytree Learning can be solved in single-exponential FPT time for the number of variables. Moreover, we consider the influence of d, the number of variables that might receive a nonempty parent set in the final DAG on the complexity of Polytree Learning. We show that Polytree Learning is presumably not fixed-parameter tractable for d, unlike Bayesian network learning which is fixed-parameter tractable for d. Finally, we show that if d and the maximum parent set size are bounded, then we can obtain efficient algorithms.


Author(s):  
Tim A. Hartmann ◽  
Stefan Lendl ◽  
Gerhard J. Woeginger

AbstractWe study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range $$\delta >0$$ δ > 0 . In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most $$\delta $$ δ from one of these facilities. We investigate this covering problem in terms of the rational parameter $$\delta $$ δ . We prove that the problem is polynomially solvable whenever $$\delta $$ δ is a unit fraction, and that the problem is NP-hard for all non unit fractions $$\delta $$ δ . We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for $$\delta <3/2$$ δ < 3 / 2 , and it is W[2]-hard for $$\delta \ge 3/2$$ δ ≥ 3 / 2 .


2021 ◽  
Vol 68 (3) ◽  
pp. 1-40
Author(s):  
Arnab Bhattacharyya ◽  
Édouard Bonnet ◽  
László Egri ◽  
Suprovat Ghoshal ◽  
Karthik C. S. ◽  
...  

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W [1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP) , in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W [1]-hard to approximate (under randomized reductions) to some constant factor.


Algorithmica ◽  
2021 ◽  
Author(s):  
Marc Roth

AbstractWe study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and $$\#\mathsf {W[1]}$$ # W [ 1 ] -complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.


Author(s):  
Yasir Mahmood ◽  
Arne Meier

AbstractDependence Logic was introduced by Jouko Väänänen in 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parameterisations with respect to central decision problems. The model checking problem (MC) of PDL is NP-complete (Ebbing and Lohmann, SOFSEM 2012). The subject of this research is to identify a list of parameterisations (formula-size, formula-depth, treewidth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) which classically is known to be NP-complete as well (Lohmann and Vollmer, Studia Logica 2013). There we are presenting a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is FPT. Finally, we introduce a variant of the satisfiability problem, asking for a team of a given size, and show for this problem an almost complete picture.


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