Extremal probability bounds in combinatorial optimization
Extremal probability bounds in combinatorial optimization
In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are 'extremal' since they are valid across all joint distributions with the given marginals. We analyze the complexity of computing the bounds, assuming discrete marginals, and identify instances when the bounds are computable in polynomial time. For compact 0/1 V-polytopes, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. On the other hand, the tightest lower bound is shown to be strongly NP-hard to compute for compact 0/1 V-polytopes by restricting our attention to Bernoulli random variables. For compact 0/1 H-polytopes, for the special case of PERT networks arising in project management, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. The results in the paper complement existing results in the literature for computing the probability with independent random variables.
PERT, combinatorial optimization, probability bounds
2828-2858
Padmanabhan, Divya
814eb256-0645-4303-8255-1e165db4badd
Ahipasaoglu, Selin Damla
d69f1b80-5c05-4d50-82df-c13b87b02687
Ramachandra, Arjun
d7b38375-1f45-4987-94fa-953709c87bfc
Natarajan, Karthik
5045c362-74e5-417c-95b9-4556fc9f9c7e
1 December 2022
Padmanabhan, Divya
814eb256-0645-4303-8255-1e165db4badd
Ahipasaoglu, Selin Damla
d69f1b80-5c05-4d50-82df-c13b87b02687
Ramachandra, Arjun
d7b38375-1f45-4987-94fa-953709c87bfc
Natarajan, Karthik
5045c362-74e5-417c-95b9-4556fc9f9c7e
Padmanabhan, Divya, Ahipasaoglu, Selin Damla, Ramachandra, Arjun and Natarajan, Karthik
(2022)
Extremal probability bounds in combinatorial optimization.
SIAM Journal on Optimization, 32 (4), .
(doi:10.1137/21M1442504).
Abstract
In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are 'extremal' since they are valid across all joint distributions with the given marginals. We analyze the complexity of computing the bounds, assuming discrete marginals, and identify instances when the bounds are computable in polynomial time. For compact 0/1 V-polytopes, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. On the other hand, the tightest lower bound is shown to be strongly NP-hard to compute for compact 0/1 V-polytopes by restricting our attention to Bernoulli random variables. For compact 0/1 H-polytopes, for the special case of PERT networks arising in project management, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. The results in the paper complement existing results in the literature for computing the probability with independent random variables.
Text
2109.01591
- Accepted Manuscript
More information
Accepted/In Press date: 1 July 2022
e-pub ahead of print date: 17 November 2022
Published date: 1 December 2022
Keywords:
PERT, combinatorial optimization, probability bounds
Identifiers
Local EPrints ID: 473726
URI: http://eprints.soton.ac.uk/id/eprint/473726
ISSN: 1052-6234
PURE UUID: 1caa00fb-96be-43cf-bf94-bfd9451b97dc
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Date deposited: 30 Jan 2023 19:49
Last modified: 17 Mar 2024 04:03
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Contributors
Author:
Divya Padmanabhan
Author:
Arjun Ramachandra
Author:
Karthik Natarajan
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