Approximating Multistage Matching Problems

Chimani, Markus; Troost, Niklas; Wiedera, Tilo

doi:10.48550/arxiv.2002.06887

Cited by 3 publications

(7 citation statements)

References 18 publications

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“…Overall, we have S ⊆ b∈B t,j Ãt,j (b). By (7) it follows that S ⊇ b∈B t,j Ãt,j (b) as well, thus S = b∈B t,j Ãt,j (b). I.e, Ãt,j is an assignment of S.…”

Section: Reductionmentioning

confidence: 93%

“…Many optimization problem were considered under this framework. These include matching [2,7], clustering [8], subset sum [3], vertex cover [13] and minimum s − t path [14].…”

Section: Related Workmentioning

confidence: 99%

“…The assignment Ãt,j is a feasible assignment w.r.t Kt,j since for each bin b ∈ B t,j it holds that (7). If i ∈ S t then t / ∈ D and thus (i, D) ∈ Ãt,j (b * ).…”

Section: Reductionmentioning

confidence: 99%

“…The assignment Ãt,j is a feasible assignment w.r.t Kt,j since for each bin b ∈ B t,j it holds that Let (i, D) ∈ S. If i ∈ S t there is b ∈ B t,j such that i ∈ A t,j (b), hence (i, D) ∈ Ãt,j (b) by (7). If i ∈ S t then t / ∈ D and thus (i, D) ∈ Ãt,j (b * ).…”

Section: B1 Reductionmentioning

confidence: 99%

See 3 more Smart Citations

General Knapsack Problems in a Dynamic Setting

Fairstein¹,

Kulik²,

Joseph³

et al. 2021

Preprint

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The world is dynamic and changes over time, thus any optimization problems used to model real life problems must address this dynamic nature. Otherwise, ignoring the dynamicity of the problem will result in locally optimal solutions on the one hand, but inadequate on the other hand. The multistage model was introduced with this goal in mind. In the multistage model we are given a series of instance of an optimization problem, and a solution is provided for each instance. The strive for continuous and similar solutions over time are quantified and integrated into the objective function. In this paper we consider the Generalized Multistage d-Knapsack problem, a generalization of the multistage variants of the Multiple Knapsack problem as well as the d-Dimensional Knapsack problem. We present a PTAS for Generalized Multistage d-Knapsack.

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“…Overall, we have S ⊆ b∈B t,j Ãt,j (b). By (7) it follows that S ⊇ b∈B t,j Ãt,j (b) as well, thus S = b∈B t,j Ãt,j (b). I.e, Ãt,j is an assignment of S.…”

Section: Reductionmentioning

confidence: 93%

“…Many optimization problem were considered under this framework. These include matching [2,7], clustering [8], subset sum [3], vertex cover [13] and minimum s − t path [14].…”

Section: Related Workmentioning

confidence: 99%

“…The assignment Ãt,j is a feasible assignment w.r.t Kt,j since for each bin b ∈ B t,j it holds that (7). If i ∈ S t then t / ∈ D and thus (i, D) ∈ Ãt,j (b * ).…”

Section: Reductionmentioning

confidence: 99%

Section: B1 Reductionmentioning

confidence: 99%

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General Knapsack Problems in a Dynamic Setting

Fairstein¹,

Kulik²,

Joseph³

et al. 2021

Preprint

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show abstract

“…Perfect Matching (Section 5). In the multistage setting, Perfect Matching is among the problems most intensively studied [3,4,13,26,39]. Given a sequence of graphs (G i ) τ i=1 and an integer ℓ, Diverse Multistage Perfect Matching asks whether there is a sequence (M i ) τ i=1 such that each M i is a perfect matching in G i , and |M i ∆M i+1 | ≥ ℓ for all i ∈ [τ − 1].…”

Section: Inputmentioning

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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A Faster Parameterized Algorithm for Temporal Matching

Zschoche¹

2020

Preprint

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A temporal graph is a sequence of graphs (called layers) over the same vertex set-describing a graph topology which is subject to discrete changes over time. A ∆-temporal matching M is a set of time edges (e, t) (an edge e paired up with a point in time t) such that for all distinct time edges (e, t), (e ′ , t ′ ) ∈ M we have that e and e ′ do not share an endpoint, or the time-labels t and t ′ are at least ∆ time units apart. Mertzios et al. [STACS '20] provided a 2 O(∆ν) • |G| O(1) -time algorithm to compute the maximum size of ∆-temporal matching in a temporal graph G, where |G| denotes the size of G, and ν is the ∆-vertex cover number of G. The ∆-vertex cover number is the minimum number of vertices which are needed to hit (or cover) all edges in any ∆ consecutive layers of the temporal graph. We show an improved algorithm to compute a ∆-temporal matching of maximum size with a running time of ∆ O(ν) • |G| and hence provide an exponential speedup in terms of ∆.

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Approximating Multistage Matching Problems

Cited by 3 publications

References 18 publications

General Knapsack Problems in a Dynamic Setting

General Knapsack Problems in a Dynamic Setting

Parameterized Algorithms for Diverse Multistage Problems

A Faster Parameterized Algorithm for Temporal Matching

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