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Preprints, Working Papers, ... (Working Paper) Year : 2023

Independent set reconfiguration: general and RNA-focused parameterized algorithms

Abstract

In this paper, we study the Independent Set (IS) reconfiguration problem in graphs, and its applications to RNA kinetics. An IS reconfiguration is a scenario transforming an IS $L$ into another IS $R$, inserting/removing one vertex at a time while keeping the cardinalities of intermediate sets as large as possible. We focus on the \emph{bipartite} variant where only start and end vertices are allowed in intermediate ISs. Our motivation is an application to the \emph{RNA energy barrier}, a classic hard problem from bioinformatics, which asks, given two RNA structures given as input, whether there exists a reconfiguration pathway connecting them and staying below an energy threshold. A natural parameter for this problem would be the difference between the initial IS size and the threshold (\emph{barrier}). We first show the para-NP hardness of the problem with respect to this parameter. We then investigate two new parameters, the \emph{cardinality range} $\range$ and a measure of \emph{arboricity} $\Phi$. $\rho$ denotes the maximum allowed size difference between an IS along the reconfiguration and a maximum IS, while $\Phi$ is a measure of the amount of ``branching'' in the two input RNA structures. We show that bipartite IS reconfiguration is XP for $\range$ in the general case, and XP for $\Phi$ in the sub-case of bipartite graphs stemming from RNA instances. %, and FPT when combining both $\Phi$ and $\range$ in the RNA case. We give two different routes yielding XP algorithms for $\rho$: The first is a direct $O(n^{2})$-space, $O(n^{2\rho+2.5})$-time algorithm based on a separation lemma; The second builds on a parameterized equivalence with the directed pathwidth problem, leading to a $O(n^{\rho+1})$-space, $O(n^{\rho+2})$-time algorithm for the reconfiguration problem through an adaptation of a prior result by Tamaki \cite{Tamaki2011}. This equivalence is an interesting result in its own right, connecting a reconfiguration problem (which is essentially a \emph{connectivity} problem within a \emph{reconfiguration network}) with a \emph{structural} parameter for an auxiliary graph. For $\Phi$, our $O(n^{\Phi+1})$-algorithm stems from seeing the problem as an instance of \emph{minimum cumulative-cost scheduling}, and relies on a \emph{merging} procedure that might be of independent interest. These results improve upon a partial $O(n^{2\rho+2.5})$-algorithm that only applied to the RNA case. We also demonstrate their practicality of these algorithms through a benchmark on small random RNA instances.
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Dates and versions

hal-04094405 , version 1 (11-05-2023)

Identifiers

  • HAL Id : hal-04094405 , version 1

Cite

Théo Boury, Laurent Bulteau, Bertrand Marchand, Yann Ponty. Independent set reconfiguration: general and RNA-focused parameterized algorithms. 2023. ⟨hal-04094405⟩
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