Faster FPTASes for Counting and Random Generation of Knapsack Solutions

Rizzi, Roméo; Tomescu, Alexandru I.

doi:10.1007/978-3-662-44777-2_63

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…, s m , k are also computed in binary). We can then apply an FPTAS to this #Knapsack instance, as shown in [18,29].…”

Section: Appendix C Proof Of Propositionmentioning

confidence: 99%

Model Interpretability through the Lens of Computational Complexity

Barceló,

Monet,

Pérez

et al. 2020

Preprint

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In spite of several claims stating that some models are more interpretable than others -e.g., "linear models are more interpretable than deep neural networks" -we still lack a principled notion of interpretability to formally compare among different classes of models. We make a step towards such a notion by studying whether folklore interpretability claims have a correlate in terms of computational complexity theory. We focus on local post-hoc explainability queries that, intuitively, attempt to answer why individual inputs are classified in a certain way by a given model. In a nutshell, we say that a class C 1 of models is more interpretable than another class C 2 , if the computational complexity of answering post-hoc queries for models in C 2 is higher than for those in C 1 . We prove that this notion provides a good theoretical counterpart to current beliefs on the interpretability of models; in particular, we show that under our definition and assuming standard complexity-theoretical assumptions (such as P = NP), both linear and tree-based models are strictly more interpretable than neural networks. Our complexity analysis, however, does not provide a clear-cut difference between linear and tree-based models, as we obtain different results depending on the particular post-hoc explanations considered. Finally, by applying a finer complexity analysis based on parameterized complexity, we are able to prove a theoretical result suggesting that shallow neural networks are more interpretable than deeper ones.

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“…, s m , k are also computed in binary). We can then apply an FPTAS to this #Knapsack instance, as shown in [18,29].…”

Section: Appendix C Proof Of Propositionmentioning

confidence: 99%

Model Interpretability through the Lens of Computational Complexity

Barceló,

Monet,

Pérez

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…the number of solutions that do not violate the capacity constraint, is #P-complete. As a consequence, different counting approaches have been introduced to approximately count the number of feasible solutions [4,18,22].…”

Section: Introductionmentioning

confidence: 99%

Exact Counting and Sampling of Optima for the Knapsack Problem

Bossek¹,

Neumann²,

Neumann³

2021

Preprint

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Computing sets of high quality solutions has gained increasing interest in recent years. In this paper, we investigate how to obtain sets of optimal solutions for the classical knapsack problem. We present an algorithm to count exactly the number of optima to a zero-one knapsack problem instance. In addition, we show how to efficiently sample uniformly at random from the set of all global optima. In our experimental study, we investigate how the number of optima develops for classical random benchmark instances dependent on their generator parameters. We find that the number of global optima can increase exponentially for practically relevant classes of instances with correlated weights and profits which poses a justification for the considered exact counting problem.

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“…A closely related problem is #Knapsack, which asks for the number of subsets S such that s∈S s ≤ t. There are extensive studies on approximation algorithms for the #Knapsack problem [6,8,13,7]. Our algorithm can solve the modulo p version # p Knapsack in near-linear pseudopolynomial time for prime p > t. 17:2 A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum Compared to the previous near-linear time algorithm for Subset Sum by Bringmann [4], our algorithm is simpler and more practical.…”

Section: Introductionmentioning

confidence: 99%

A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum

Jin,

2018

Preprint

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Given a multiset S of n positive integers and a target integer t, the Subset Sum problem asks to determine whether there exists a subset of S that sums up to t. The current best deterministic algorithm, by Koiliaris and Xu [SODA'17], runs in Õ( √ nt) time, where Õ hides poly-logarithm factors. Bringmann [SODA'17] later gave a randomized Õ(n + t) time algorithm using two-stage color-coding. The Õ(n + t) running time is believed to be near-optimal.In this paper, we present a simple and elegant randomized algorithm for Subset Sum in Õ(n + t) time. Our new algorithm actually solves its counting version modulo prime p > t, by manipulating generating functions using FFT. ACM Subject Classification Theory of computation → Algorithm design techniquesKeywords and phrases subset sum, formal power series, FFT Acknowledgements The authors would like to thank the anonymous reviewers for their helpful comments.

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Faster FPTASes for Counting and Random Generation of Knapsack Solutions

Cited by 5 publications

References 11 publications

Model Interpretability through the Lens of Computational Complexity

Model Interpretability through the Lens of Computational Complexity

Exact Counting and Sampling of Optima for the Knapsack Problem

A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum

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