Multistage graph problems on a global budget

Heeger, Klaus; Himmel, Anne-Sophie; Kammer, Frank; Niedermeier, Rolf; Renken, Malte; Sajenko, Andrej

doi:10.1016/j.tcs.2021.04.002

Cited by 18 publications

(13 citation statements)

References 12 publications

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“…Here, a solution to the i-th instance should be sufficiently different from the δ previous solutions in the sequence; our work covers the case δ = 1. In some multistage scenarios a "global view" [27] on the symmetric differences is desired. In context of this paper this means that two consecutive solutions can have a small symmetric difference as long as the sum of all consecutive symmetric differences is at least ℓ.…”

Section: Discussionmentioning

confidence: 99%

Parameterized Algorithms for Diverse Multistage Problems

Kellerhals,

Renken,

Zschoche

2021

Preprint

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The world is rarely static -many problems need not only be solved once but repeatedly, under changing conditions. This setting is addressed by the "multistage" view on computational problems. We study the "diverse multistage" variant, where consecutive solutions of large variety are preferable to similar ones, e.g. for reasons of fairness or wear minimization. While some aspects of this model have been tackled before, we introduce a framework allowing us to prove that a number of diverse multistage problems are fixed-parameter tractable by diversity, namely Perfect Matching, s-t Path, Matroid Independent Set, and Plurality Voting. This is achieved by first solving special, colored variants of these problems, which might also be of independent interest.

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Section: Discussionmentioning

confidence: 99%

Parameterized Algorithms for Diverse Multistage Problems

Kellerhals,

Renken,

Zschoche

2021

Preprint

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“…The solution may only be changed by a certain amount between any two consecutive stages. Heeger et al [22] started the parameterized research of multistage graph problems on a global budget where there is no restriction on the number of changes between any two consecutive layers, but instead a restriction on the total number of changes made throughout the lifetime of the instance. ) and an integer D ∈ N 0 .…”

Section: Global Budgetmentioning

confidence: 99%

“…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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“…One particular flavor of temporal graph problems is concerned with obtaining a sequence of solutions-one for each stage-while optimizing a global quantity. These problems are often referred to as multistage problems and gained much attention in recent years [2-5, 13, 15, 16], including the realm of matchings: e.g., the authors of [16] show W [1]-hardness for finding a set of at least k edges whose intersection with each stage is a matching.…”

Section: Related Workmentioning

confidence: 99%

Approximating Multistage Matching Problems

Chimani¹,

Troost²,

Wiedera³

2020

Preprint

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In multistage perfect matching problems we are given a sequence of graphs on the same vertex set and asked to find a sequence of perfect matchings, corresponding to the sequence of graphs, such that consecutive matchings are as similar as possible. More precisely, we aim to maximize the intersections, or minimize the unions between consecutive matchings. We show that these problems are NP-hard even in very restricted scenarios.We propose new approximation algorithms and present methods to transfer results between different problem variants without loosing approximation guarantees.

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Multistage graph problems on a global budget

Cited by 18 publications

References 12 publications

Parameterized Algorithms for Diverse Multistage Problems

Parameterized Algorithms for Diverse Multistage Problems

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

Approximating Multistage Matching Problems

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