Change-making problems revisited: a parameterized point of view

Goebbels, Steffen; Gurski, Frank; Rethmann, Jochen; Yilmaz, Eda

doi:10.1007/s10878-017-0143-z

Cited by 8 publications

(10 citation statements)

References 21 publications

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“…One aspect of the algorithm above is reducing the quadratic objective function. The standard approach, also used in kernelization of weighted problems [1,7,13,19,21,42,43] is to use a theorem of Frank and Tardos [16] which "kernelizes" a linear objective function if the dimension is a parameter. However, we deal with (1) a quadratic convex (non-linear) function, (2) over a space of large dimension.…”

Section: Configuration Lpmentioning

confidence: 99%

Scheduling Kernels via Configuration LP

Knop¹,

Koutecký²

2020

Preprint

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Makespan minimization (on parallel identical or unrelated machines) is arguably the most natural and studied scheduling problem. A common approach in practical algorithm design is to reduce the size of a given instance by a fast preprocessing step while being able to recover key information even after this reduction. This notion is formally studied as kernelization (or simply, kernel) -a polynomial time procedure which yields an equivalent instance whose size is bounded in terms of some given parameter. It follows from known results that makespan minimization parameterized by the longest job processing time pmax has a kernelization yielding a reduced instance whose size is exponential in pmax. Can this be reduced to polynomial in pmax?We answer this affirmatively not only for makespan minimization, but also for the (more complicated) objective of minimizing the weighted sum of completion times, also in the setting of unrelated machines when the number of machine kinds is a parameter.Our algorithm first solves the Configuration LP and based on its solution constructs a solution of an intermediate problem, called huge N -fold integer programming. This solution is further reduced in size by a series of steps, until its encoding length is polynomial in the parameters. Then, we show that huge N -fold IP is in NP, which implies that there is a polynomial reduction back to our scheduling problem, yielding a kernel.Our technique is highly novel in the context of kernelization, and our structural theorem about the Configuration LP is of independent interest. Moreover, we show a polynomial kernel for huge N -fold IP conditional on whether the so-called separation subproblem can be solved in polynomial time. Considering that integer programming does not admit polynomial kernels except for quite restricted cases, our "conditional kernel" provides new insight.

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Section: Configuration Lpmentioning

confidence: 99%

Scheduling Kernels via Configuration LP

Knop¹,

Koutecký²

2020

Preprint

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“…The change-making problem has some practical applications, such as network design [6], cutting-stock, and capital allocation [11]. From a theoretical perspective, Magazine, Nemhauser, and Trotter Jr. [10] analyzed conditions for the knapsack problem to be solvable with a greedy method, Tien and Hu [13] studied the gap between greedy and optimal solutions of the change-making problem, Adamaszek and Adamaszek [1] revealed some relationships between canonical systems and their subsystems, Goebbels, Gurski, Rethmann, and Yilmaz [7] considered approximation algorithms and fixed-parameter tractability for the changemaking problem, and Chan and He [3] recently proposed faster dynamic programming-based algorithms for the change-making and related problems.…”

Section: Related Results On Characterizationmentioning

confidence: 99%

Characterization of canonical systems with six types of coins for the change-making problem

Suzuki¹,

Miyashiro²

2021

Preprint

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This paper analyzes a necessary and sufficient condition for the change-making problem to be solvable with a greedy algorithm. The change-making problem is to minimize the number of coins used to pay a given value in a specified currency system. This problem is NP-hard, and therefore the greedy algorithm does not always yield an optimal solution. Yet for almost all real currency systems, the greedy algorithm outputs an optimal solution. A currency system for which the greedy algorithm returns an optimal solution for any value of payment is called a canonical system. Canonical systems with at most five types of coins have been characterized in previous studies. In this paper, we give characterization of canonical systems with six types of coins, and we propose a partial generalization of characterization of canonical systems.

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“…The standard approach, also used in kernelization of weighted problems [6,8,9,16,25,33,37] is to use the above proposition which "kernelizes" a linear objective function if the dimension is bounded by a parameter.…”

Section: Reducing Numbers In the Inputmentioning

confidence: 99%

On Polynomial Kernels for Traveling Salesperson Problem and its Generalizations

Blažej¹,

Choudhary²,

Knop³

et al. 2022

Preprint

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For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today's computation, we employ one of the most successful models of such precomputation-the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations.We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of Subset TSP, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.

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Change-making problems revisited: a parameterized point of view

Cited by 8 publications

References 21 publications

Scheduling Kernels via Configuration LP

Scheduling Kernels via Configuration LP

Characterization of canonical systems with six types of coins for the change-making problem

On Polynomial Kernels for Traveling Salesperson Problem and its Generalizations

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