Huge Unimodular $n$-Fold Programs

Onn, Shmuel; Sarrabezolles, Pauline

doi:10.1137/151004227

Cited by 4 publications

(9 citation statements)

References 12 publications

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“…, n Θ with n = n 1 + · · · + n Θ , such that n j denotes the number of jobs of type j that are to be scheduled. Modeling our terminology after Onn [29], we call a problem huge when the numbers n j on input are given in binary. For example, the Cutting Stock problem can be seen as the huge variant of the Bin Packing problem.…”

Section: Our Contributionmentioning

confidence: 99%

Scheduling meets n-fold integer programming

Knop

Koutecký

2017

J Sched

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Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In 2014, Mnich and Wiese initiated a systematic study in this direction. In this paper we continue this study and show that several additional cases of fundamental scheduling problems are fixed parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which we believe to be highly relevant for the parameterized complexity community. This paper serves to showcase and highlight this technique. Specifically, we show the following four scheduling problems to be fixed-parameter tractable, where pmax is the maximum processing time of a job and wmax is the maximum weight of a job:-Makespan minimization on uniformly related machines (Q||Cmax) parameterized by pmax, -Makespan minimization on unrelated machines (R||Cmax) parameterized by pmax and the number of kinds of machines (defined later), -Sum of weighted completion times minimization on unrelated machines (R|| wjCj) parameterized by pmax + wmax and the number of kinds of machines, -The same problem, R|| wjCj, parameterized by the number of distinct job times and the number of machines.

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Section: Our Contributionmentioning

confidence: 99%

Scheduling meets n-fold integer programming

Knop

Koutecký

2017

J Sched

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“…In particular, what is the complexity of deciding the existence of a huge four-way k × l × m × n table with given sums, with k, l, m fixed and n encoded in binary, with t types? It is known that for fixed t the problem is in P, and for variable t it is in NP intersect coNP but is not known to be in P even for 3 × 3 × 3 × n tables, see [17]. We also do not know whether the problem, with k, l, m as parameters, with variable or even fixed t, is fixed-parameter tractable.…”

Section: Open Problemsmentioning

confidence: 99%

“…, x n ) consisting of n many l × m layers. Following [17], call the problem huge if the variable number n of layers is encoded in binary. We are then given t types of layers, where each type k has its column sums vector u k ∈ Z m + and row sums vector v k ∈ Z l + .…”

Section: Unimodular Integer Carathéodorymentioning

confidence: 99%

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Huge tables and multicommodity flows are fixed-parameter tractable via unimodular integer Carathéodory

Onn

2017

Journal of Computer and System Sciences

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The three-way table problem is to decide if there exists an l × m × n table satisfying given line sums, and find a table if yes. Recently, it was shown to be fixed-parameter tractable with parameters l, m. Here we extend this and show that the huge version of the problem, where the variable side n is encoded in binary, is also fixed-parameter tractable with parameters l, m. We also conclude that the huge multicommodity flow problem with a huge number of consumers is fixed-parameter tractable. One of our tools is a theorem about unimodular monoids which is of interest on its own right.

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“…As for applications, n-fold ILPs are broadly used to model various problems such as string, flow or scheduling problems. We refer to the works [5,10,11,12,15,19] and the references therein for an overview.…”

Section: Related Workmentioning

confidence: 99%

Near-Linear Time Algorithm for $n$-Fold ILPs via Color Coding

Jansen¹,

Lassota²,

Rohwedder³

2020

SIAM J. Discrete Math.

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We study an important case of ILPs max{c T x | Ax = b, l ≤ x ≤ u, x ∈ Z nt } with n • t variables and lower and upper bounds , u ∈ Z nt. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s × t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and ∆, where ∆ is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Ω(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs∆) O(r 2 s+s 2) L 2 • nt log O(1) (nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L.

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Huge Unimodular $n$-Fold Programs

Cited by 4 publications

References 12 publications

Scheduling meets n-fold integer programming

Scheduling meets n-fold integer programming

Huge tables and multicommodity flows are fixed-parameter tractable via unimodular integer Carathéodory

Near-Linear Time Algorithm for $n$-Fold ILPs via Color Coding

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