Parameterized Algorithms for Diverse Multistage Problems

Kellerhals, Leon; Renken, Malte; Zschoche, Philipp

doi:10.4230/lipics.esa.2021.55

Cited by 4 publications

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“…, I τ ) of instances of some problem P as input, and it asks for a "robust" sequence of solutions to the instances in the sense that any two consecutive solutions are similar. Several classical problems have been studied in the multistage model, both from an approximate [1][2][3][4] and from a parameterized [8,19,20,23,31,34] algorithmics point of view. While E E-MstP and V V-MstP adhere to the original multistage model, our two problems E∩E-MstP and V∩V-MstP can be seen as a novel and natural variation of the multistage model by replacing the goal of consecutive similarity with consecutive dissimilarity.…”

Section: Introductionmentioning

confidence: 99%

Multistage s–t Path: Confronting Similarity with Dissimilarity

et al. 2023

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Addressing a quest by Gupta et al. (in: Proceedings of the 41st international colloquium on automata, languages, and programming (ICALP 2014), vol 8572 of LNCS. Springer, pp 563–575, 2014), we provide a first, comprehensive study of finding a short s–t path in the multistage graph model, referred to as the Multistages–tPath problem. Herein, given a sequence of graphs over the same vertex set but changing edge sets, the task is to find short s–t paths in each graph (“snapshot”) such that in the found path sequence the consecutive s–t paths are “similar”. We measure similarity by the size of the symmetric difference of either the vertex set (vertex-similarity) or the edge set (edge-similarity) of any two consecutive paths. We prove that these two variants of Multistages–tPath are already $${\text {NP}}$$ NP -hard for an input sequence of only two snapshots and maximum vertex degree four. Motivated by this fact and natural applications of this scenario e.g. in traffic route planning, we perform a parameterized complexity analysis. Among other results, for both variants, vertex- and edge-similarity, we prove parameterized hardness ($${\text {W[1]}}$$ W[1] -hardness) regarding the parameter path length (solution size). As a further conceptual investigation, we then modify the multistage model by asking for dissimilar consecutive paths. As one of the main technical results (employing so-called representative sets known from non-temporal settings), we prove that dissimilarity allows for fixed-parameter tractability for the parameter solution size, contrasting with our W[1]-hardness proof of the corresponding similarity case. We also provide partially positive results concerning efficient and effective data reduction (kernelization).

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Section: Introductionmentioning

confidence: 99%

Multistage s–t Path: Confronting Similarity with Dissimilarity

et al. 2023

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“…The multistage framework is still young, but several problems have been investigated in it, mostly in the last couple of years, including Matching [3,9,21], Knapsack [4], s-t Path [20], Vertex Cover [19], Committee Election [7], and others [2]. The framework has also been extended to goals other than minimizing the number of changes in the solution between layers [22,25]. Since these types of problems are NP-hard even in fairly restricted settings, most research has focused on their parameterized complexity and approximability.…”

Section: Introductionmentioning

confidence: 99%

Bipartite Temporal Graphs and the Parameterized Complexity of Multistage 2-Coloring

Fluschnik¹,

Kunz²

2021

Preprint

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We consider the algorithmic complexity of recognizing bipartite temporal graphs. Rather than defining these graphs solely by their underlying graph or individual layers, we define a bipartite temporal graph as one in which every layer can be 2-colored in a way that results in few changes between any two consecutive layers. This approach follows the framework of multistage problems that has received a growing amount of attention in recent years. We investigate the complexity of recognizing these graphs. We show that this problem is NP-hard even if there are only two layers or if only one change is allowed between consecutive layers. We consider the parameterized complexity of the problem with respect to several structural graph parameters, which we transfer from the static to the temporal setting in three different ways. Finally, we consider a version of the problem in which we only restrict the total number of changes throughout the lifetime of the graph. We show that this variant is fixed-parameter tractable with respect to the number of changes. * Supported by the DFG, project MATE (NI 369/19). † Supported by the DFG Research Training Group 2434 "Facets of Complexity".

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Multistage Vertex Cover

et al. 2022

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The NP-complete Vertex Cover problem asks to cover all edges of a graph by a small (given) number of vertices. It is among the most prominent graph-algorithmic problems. Following a recent trend in studying temporal graphs (a sequence of graphs, so-called layers, over the same vertex set but, over time, changing edge sets), we initiate the study of Multistage Vertex Cover. Herein, given a temporal graph, the goal is to find for each layer of the temporal graph a small vertex cover and to guarantee that two vertex cover sets of every two consecutive layers differ not too much (specified by a given parameter). We show that, different from classic Vertex Cover and some other dynamic or temporal variants of it, Multistage Vertex Cover is computationally hard even in fairly restricted settings. On the positive side, however, we also spot several fixed-parameter tractability results based on some of themost natural parameterizations.

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Parameterized Algorithms for Diverse Multistage Problems

Cited by 4 publications

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Multistage s–t Path: Confronting Similarity with Dissimilarity

Multistage s–t Path: Confronting Similarity with Dissimilarity

Bipartite Temporal Graphs and the Parameterized Complexity of Multistage 2-Coloring

Multistage Vertex Cover

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