Multiplicative Parameterization Above a Guarantee

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.

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On the complexity of Gödel's proof predicate

Journal of Symbolic Logic ◽

10.2178/jsl/1264433918 ◽

2010 ◽

Vol 75 (1) ◽

pp. 239-254 ◽

Cited By ~ 9

Author(s):

Yijia Chen ◽

Jörg Flum

Keyword(s):

Classical Problem ◽

Order Logic ◽

First Order Logic ◽

Fixed Parameter Tractability ◽

Fixed Parameter Tractable ◽

First Order ◽

Fixed Parameter ◽

Parameterized Problem ◽

Computable Bound ◽

Increasing Functions

AbstractThe undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natural number n in unary as input and a first-order sentence φ as parameter, asks whether φ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-Gödel. We show that p-Gödel is not fixed-parameter tractable if DTIME(hO(1)) ≠ NTIME(hO(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the construction problem associated with p-Gödel.

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Oritatami: A Computational Model for Molecular Co-Transcriptional Folding

International Journal of Molecular Sciences ◽

10.3390/ijms20092259 ◽

2019 ◽

Vol 20 (9) ◽

pp. 2259 ◽

Cited By ~ 5

Author(s):

Cody Geary ◽

Pierre-Étienne Meunier ◽

Nicolas Schabanel ◽

Shinnosuke Seki

Keyword(s):

Local Environment ◽

Fixed Parameter Tractable ◽

Synthetic Dna ◽

Local Environments ◽

Fixed Parameter ◽

Fixed Constant ◽

Different Shapes ◽

Recent Experimental Work ◽

Single Sequence ◽

Rna Fold

We introduce and study the computational power of Oritatami, a theoretical model that explores greedy molecular folding, whereby a molecular strand begins to fold before its production is complete. This model is inspired by our recent experimental work demonstrating the construction of shapes at the nanoscale from RNA, where strands of RNA fold into programmable shapes during their transcription from an engineered sequence of synthetic DNA. In the model of Oritatami, we explore the process of folding a single-strand bit by bit in such a way that the final fold emerges as a space-time diagram of computation. One major requirement in order to compute within this model is the ability to program a single sequence to fold into different shapes dependent on the state of the surrounding inputs. Another challenge is to embed all of the computing components within a contiguous strand, and in such a way that different fold patterns of the same strand perform different functions of computation. Here, we introduce general design techniques to solve these challenges in the Oritatami model. Our main result in this direction is the demonstration of a periodic Oritatami system that folds upon itself algorithmically into a prescribed set of shapes, depending on its current local environment, and whose final folding displays the sequence of binary integers from 0 to N = 2 k − 1 with a seed of size O ( k ) . We prove that designing Oritatami is NP-hard in the number of possible local environments for the folding. Nevertheless, we provide an efficient algorithm, linear in the length of the sequence, that solves the Oritatami design problem when the number of local environments is a small fixed constant. This shows that this problem is in fact fixed parameter tractable (FPT) and can thus be solved in practice efficiently. We hope that the numerous structural strategies employed in Oritatami enabling computation will inspire new architectures for computing in RNA that take advantage of the rapid kinetic-folding of RNA.

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Soft Constraints of Difference and Equality

Journal of Artificial Intelligence Research ◽

10.1613/jair.3197 ◽

2011 ◽

Vol 41 ◽

pp. 97-130 ◽

Cited By ~ 2

Author(s):

E. Hebrard ◽

D. Marx ◽

B. O'Sullivan ◽

I. Razgon

Keyword(s):

Linear Time ◽

Combinatorial Problems ◽

Global Constraints ◽

Fixed Parameter Tractable ◽

Common Value ◽

Arc Consistency ◽

Standard Cost ◽

Fixed Parameter ◽

Difference And Equality ◽

Np Hardness

In many combinatorial problems one may need to model the diversity or similarity of assignments in a solution. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type, we can use soft variants of the well known AllDifferent and AllEqual constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however achieving bounds consistency can be done in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor 1/2 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.

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Complexidade Parametrizada de Cliques e Conjuntos Independentes em Grafos Prismas Complementares

10.5753/etc.2018.3166 ◽

2018 ◽

Author(s):

Priscila Camargo ◽

Alan D. A. Carneiro ◽

Uéverton S. Santos

Keyword(s):

Disjoint Union ◽

Perfect Matching ◽

Independent Set ◽

Point Of View ◽

Polynomial Kernel ◽

Input Graph ◽

Fixed Parameter Tractable ◽

Fixed Parameter ◽

Np Complete ◽

Complementary Prism

The complementary prism GG¯ arises from the disjoint union of the graph G and its complement G¯ by adding the edges of a perfect matching joining pairs of corresponding vertices of G and G¯. The classical problems of graph theory, clique and independent set were proved NP-complete when the input graph is a complemantary prism. In this work, we study the complexity of both problems in complementary prisms graphs from the parameterized complexity point of view. First, we prove that these problems have a kernel and therefore are Fixed-Parameter Tractable (FPT). Then, we show that both problems do not admit polynomial kernel.

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Metric Dimension Parameterized By Treewidth

Algorithmica ◽

10.1007/s00453-021-00808-9 ◽

2021 ◽

Author(s):

Édouard Bonnet ◽

Nidhi Purohit

Keyword(s):

Computable Function ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Minimum Size ◽

Metric Dimension ◽

Resolving Set ◽

Input Graph ◽

Outerplanar Graphs ◽

Bounded Treewidth ◽

Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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