Scheduling meets n-fold integer programming

Knop, Dušan; Koutecký, Martin

doi:10.1007/s10951-017-0550-0

Cited by 59 publications

(89 citation statements)

References 31 publications

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“…In 2013, Hemmecke, Onn and Romanchuk presented an FPT algorithm for n-fold integer programming with the running time of f (s 1 , s 2 , ||A|| ∞ )n 3 L where f is some computable function, ||A|| ∞ is the largest absolute value among all entries of A and L is the encoding length of the problem. This algorithm implies an FPT algorithm parameterized by the largest job processing time for P ||C max and many other scheduling problems [13]. We further extend their result by considering a broader class of integer programming, namely tree-fold integer programming as we describe as follows.…”

Section: Introductionmentioning

confidence: 89%

“…In 2013, Mnich and Wiese [17] provided an FPT (fixed parameter tractable) algorithm parameterized by the largest job processing time w max = max{w j |1 ≤ j ≤ n}. Very recently, Knop and Koutecký [13] observes that many scheduling problems can be formulated as an integer program with a special structure called n-fold integer program. Exploiting the FPT algorithm for the n-fold integer program [5] they are able to show an FPT algorithm for various scheduling problems.…”

Section: Introductionmentioning

confidence: 99%

“…The n-fold integer programming has received many studies in the literature. Indeed, the natural ILP formulation of the scheduling and bin packing problem falls into an n-fold integer programming, as is observed by Knop and Koutecký [13]. In 2013, Hemmecke, Onn and Romanchuk presented an FPT algorithm for n-fold integer programming with the running time of f (s 1 , s 2 , ||A|| ∞ )n 3 L where f is some computable function, ||A|| ∞ is the largest absolute value among all entries of A and L is the encoding length of the problem.…”

Section: Introductionmentioning

confidence: 99%

“…In terms of FPT algorithms, Mnich and Wiese [17] showed that P ||C max is FPT parameterized by the largest job processing time (edge weight). Very recently, Knop and Koutecký [13] observes the relationship between the scheduling problem and n-fold integer programming in terms of FPT algorithms. Indeed, they show that a variety of scheduling problems, including P ||C max , could be formulated as an n-fold integer programming.…”

Section: Introductionmentioning

confidence: 99%

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Covering a tree with rooted subtrees – parameterized and approximation algorithms

Chen¹,

Marx²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the multiple traveling salesman problem on a weighted tree. In this problem there are m salesmen located at the root initially. Each of them will visit a subset of vertices and return to the root. The goal is to assign a tour to every salesman such that every vertex is visited and the longest tour among all salesmen is minimized. The problem is equivalent to the subtree cover problem, in which we cover a tree with rooted subtrees such that the weight of the maximum weighted subtree is minimized. The classical machine scheduling problem can be viewed as a special case of our problem when the given tree is a star. We provide approximation and parameterized algorithms for this problem. We first present a PTAS (Polynomial Time Approximation Scheme). We then observe that, the problem remains NP-hard even if tree height and edge weight are constant, and present an FPT algorithm for this problem parameterized by the largest tour length. To achieve the FPT algorithm, we first formulate the problem as an integer linear program having a certain "tree-fold" structure. Then we show that an ILP with such a structure is FPT, which is a generalization of an earlier FPT result for n-fold integer programming by Hemmecke, Onn and Romanchuk [5]. This extension of n-fold ILP may be of independent interest. Keywords: Approximation schemes; Fixed Parameter Tractable; Integer Programming; Scheduling IntroductionWe consider the multiple traveling salesmen problem on a given tree T = (V, E). In this problem there is a root r ∈ V where all the m salesmen are initially located. There is a weight w e ∈ Z + associated with each edge e ∈ E, which is the time consumed by a salesman if he passes this edge. Each salesman starts at r, travels a subset of the vertices and returns to r. The goal is to determine the tours traveled by each salesman such that every vertex is visited by some salesman, and the makespan, i.e., the time when the last salesman returns to r, is minimized.We observe that the tour of every salesman is actually a subtree rooted at r, and the total traveling time of each salesman is exactly twice the total weight

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Covering a tree with rooted subtrees – parameterized and approximation algorithms

Chen¹,

Marx²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Hemmecke, Onn, and Romanchuk [17] prove the following. Recently, algorithmic breakthroughs in stringology [23], computational social choice [24], scheduling [6,19,22], etc., were achieved by applying this algorithm and its subsequent non-trivial improvements.…”

Section: Introductionmentioning

confidence: 99%

Evaluating and Tuning n -fold Integer Programming

Altmanová

Knop

Koutecký

2019

ACM J. Exp. Algorithmics

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In recent years, algorithmic breakthroughs in stringology, computational social choice, scheduling, etc., were achieved by applying the theory of so-called n-fold integer programming. An n-fold integer program (IP) has a highly uniform block structured constraint matrix. Hemmecke, Onn, and Romanchuk [Math. Programming, 2013] showed an algorithm with runtime ∆ O(rst+r 2 s) n 3 , where ∆ is the largest coefficient, r, s, and t are dimensions of blocks of the constraint matrix and n is the total dimension of the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients. The algorithm works by iteratively improving a feasible solution with augmenting steps, and n-fold IPs have the special property that augmenting steps are guaranteed to exist in a not-too-large neighborhood. However, this algorithm has never been implemented and evaluated.We have implemented the algorithm and learned the following along the way. The original algorithm is practically unusable, but we discover a series of improvements which make its evaluation possible. Crucially, we observe that a certain constant in the algorithm can be treated as a tuning parameter, which yields an efficient heuristic (essentially searching in a smaller-than-guaranteed neighborhood). Furthermore, the algorithm uses an overly expensive strategy to find a "best" step, while finding only an "approximately best" step is much cheaper, yet sufficient for quick convergence. Using this insight, we improve the asymptotic dependence on n from n 3 to n 2 log n.Finally, we tested the behavior of the algorithm with various values of the tuning parameter and different strategies of finding improving steps. First, we show that decreasing the tuning parameter initially leads to an increased number of iterations needed for convergence and eventually to getting stuck in local optima, as expected. However, surprisingly small values of the parameter already exhibit good behavior while significantly lowering the time the algorithm spends per single iteration. Second, our new strategy for finding "approximately best" steps wildly outperforms the original construction. ACM Subject ClassificationTheory of computation → Parameterized complexity and exact algorithms; Mathematics of computing → Solvers; Theory of computation → Discrete optimization Keywords and phrases n-fold integer programming, integer programming, analysis of algorithms, primal heuristic, local search Supplement Material https://github.com/katealtmanova/nfoldexperiment

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A Sort & Search method for multicriteria optimization problems with applications to scheduling theory

Shang

t'Kindt

2019

Multi Criteria Decision Anal

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This paper focuses on the Sort & Search method to solve a particular class of single criterion optimization problems called multiple constraints problems. This general method enables to derive exponential-time algorithms for  -hard optimization problems. In this paper, this method is further extended to the class of multicriteria multiple constraints problems for which the set of Pareto optima is enumerated. Then, the application of this new method on some multicriteria scheduling problems is proposed, leading to new exponential-time algorithms for those problems.

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Scheduling meets n-fold integer programming

Cited by 59 publications

References 31 publications

Covering a tree with rooted subtrees – parameterized and approximation algorithms

Covering a tree with rooted subtrees – parameterized and approximation algorithms

Evaluating and Tuning n -fold Integer Programming

A Sort & Search method for multicriteria optimization problems with applications to scheduling theory

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