Combinatorial optimization with interaction costs: Complexity and solvable cases

Lendl, Stefan; Ćustić, Ante; Punnen, Abraham P.

doi:10.1016/j.disopt.2019.03.004

Cited by 15 publications

(11 citation statements)

References 52 publications

(94 reference statements)

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“…One of the main contributions of [11] is the structured analysis and modelling (using DCA) of matroid base problems and their generalizations with intersection constraints. They also present some relations of the problems studied in this paper with the Combinatorial Optimization Problem with Interaction Costs (COPIC) studied in [2,13].…”

Section: Comparison To the Results Inmentioning

confidence: 99%

Matroid bases with cardinality constraints on the intersection

2021

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Given two matroids $$\mathcal {M}_{1} = (E, \mathcal {B}_{1})$$ M 1 = ( E , B 1 ) and $$\mathcal {M}_{2} = (E, \mathcal {B}_{2})$$ M 2 = ( E , B 2 ) on a common ground set E with base sets $$\mathcal {B}_1$$ B 1 and $$\mathcal {B}_2$$ B 2 , some integer $$k \in \mathbb {N}$$ k ∈ N , and two cost functions $$c_{1}, c_{2} :E \rightarrow \mathbb {R}$$ c 1 , c 2 : E → R , we consider the optimization problem to find a basis $$X \in \mathcal {B}_{1}$$ X ∈ B 1 and a basis $$Y \in \mathcal {B}_{2}$$ Y ∈ B 2 minimizing the cost $$\sum _{e\in X} c_1(e)+\sum _{e\in Y} c_2(e)$$ ∑ e ∈ X c 1 ( e ) + ∑ e ∈ Y c 2 ( e ) subject to either a lower bound constraint $$|X \cap Y| \le k$$ | X ∩ Y | ≤ k , an upper bound constraint $$|X \cap Y| \ge k$$ | X ∩ Y | ≥ k , or an equality constraint $$|X \cap Y| = k$$ | X ∩ Y | = k on the size of the intersection of the two bases X and Y. The problem with lower bound constraint turns out to be a generalization of the Recoverable Robust Matroid problem under interval uncertainty representation for which the question for a strongly polynomial-time algorithm was left as an open question in Hradovich et al. (J Comb Optim 34(2):554–573, 2017). We show that the two problems with lower and upper bound constraints on the size of the intersection can be reduced to weighted matroid intersection, and thus be solved with a strongly polynomial-time primal-dual algorithm. We also present a strongly polynomial, primal-dual algorithm that computes a minimum cost solution for every feasible size of the intersection k in one run with asymptotic running time equal to one run of Frank’s matroid intersection algorithm. Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids. We obtain a strongly polynomial time algorithm for the recoverable robust polymatroid base problem with interval uncertainties.

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Section: Comparison To the Results Inmentioning

confidence: 99%

Matroid bases with cardinality constraints on the intersection

2021

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“…The algorithm proposed in this section satisfies another interesting property, namely the vectors d k satisfy the constant value property for all k ≥ 0. This is an important property for linearizability because the set of linearizable cost matrices for combinatorial optimization problems with interaction costs can be characterized by the constant value property, under certain conditions, see [27].…”

Section: 2mentioning

confidence: 99%

The quadratic cycle cover problem: special cases and efficient bounds

Meijer

Sotirov

2020

J Comb Optim

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The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds.In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property.The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account.

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“…If the algorithms are able to find the best possible solution, that is, to explore the entire search space, a global optimum is obtained. However, in most cases this is not possible, and instead, good solutions are found by exploring only a part of the search space, and a local optimum is obtained, which in some cases may be good enough to solve the problem [25].…”

Section: Metaheuristicsmentioning

confidence: 99%

Automating Configuration of Convolutional Neural Network Hyperparameters Using Genetic Algorithm

et al. 2020

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In recent years, Convolutional Neural Networks (CNN) have been widely used for realworld applications in the field of computer vision. Their class-leading performance, however, depends heavily on the architecture used for a given problem. In most cases, the architectures are manually optimized by the researchers, a time-consuming process hard to achieve without prior knowledge of CNN. In this paper, we propose a new genetic algorithm for the optimization of the CNN architecture for a given image classification problem. This algorithm extends and refines existing research in the field, by allowing depth exploration, introducing a novel sequential crossover operator, using an incremental selective pressure schedule over evolution (favoring higher diversity in early generations) and by evaluating individual performances over the validation set with early stopping. The technique is validated in three image classification dataset, namely, CIFAR10, MNIST and Caltech256 datasets, which are widely used benchmarks for image classification algorithms. We evaluate the performance and total execution time over these datasets, and compare our results with those achieved by the best genetic methods published so far. In all cases, we achieve better results in terms of test accuracy, consistently over different datasets, while remaining in the same orders of magnitude of execution time of the fastest approaches.

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Combinatorial optimization with interaction costs: Complexity and solvable cases

Cited by 15 publications

References 52 publications

Matroid bases with cardinality constraints on the intersection

Matroid bases with cardinality constraints on the intersection

The quadratic cycle cover problem: special cases and efficient bounds

Automating Configuration of Convolutional Neural Network Hyperparameters Using Genetic Algorithm

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