The Cunningham-Geelen Method in Practice: Branch-Decompositions and Integer Programming

Margulies, Susan S.; Ma, Jing; Hicks, Illya V.

doi:10.1287/ijoc.1120.0524

Cited by 7 publications

(1 citation statement)

References 14 publications

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“…Listing all the literature here is impossible. Instead, we point to [17,36,44,49,79,88] and the references therein as excellent summaries and starting points for exploring this diverse field of research activity.…”

Section: Mixedmentioning

confidence: 99%

Complexity of optimizing over the integers

Basu¹

2021

Preprint

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In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and "complexity" in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of mixed-integer convex optimization, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition.We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities.

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Section: Mixedmentioning

confidence: 99%

Complexity of optimizing over the integers

Basu¹

2021

Preprint

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Possible and Impossible Attempts to Solve the Treewidth Problem via ILPs

Grigoriev

2020

Treewidth, Kernels, and Algorithms

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On the optimality of pseudo-polynomial algorithms for integer programming

et al. 2022

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In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given m × n matrix A and an m-vector b = (b 1 , . . . , b m ), there is a non-negative integer n-vector x such that Ax = b. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input.The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix A is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, pathwidth. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant.

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The Cunningham-Geelen Method in Practice: Branch-Decompositions and Integer Programming

Cited by 7 publications

References 14 publications

Complexity of optimizing over the integers

Complexity of optimizing over the integers

Possible and Impossible Attempts to Solve the Treewidth Problem via ILPs

On the optimality of pseudo-polynomial algorithms for integer programming

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