Budgeted uncertainty sets have been established as a major influence on uncertainty modeling for robust optimization problems. A drawback of such sets is that the budget constraint only restricts the global amount of cost increase that can be distributed by an adversary. Local restrictions, while being important for many applications, cannot be modeled this way.

We introduce a new variant of budgeted uncertainty sets, called locally budgeted uncertainty. In this setting, the uncertain parameters are partitioned, such that a classic budgeted uncertainty set applies to each part of the partition, called region.

In a theoretical analysis, we show that the robust counterpart of such problems for a constant number of regions remains solvable in polynomial time, if the underlying nominal problem can be solved in polynomial time as well. If the number of regions is unbounded, we show that the robust selection problem remains solvable in polynomial time, while also providing hardness results for other combinatorial problems.

In computational experiments using both random and real-world data, we show that using locally budgeted uncertainty sets can have considerable advantages over classic budgeted uncertainty sets.

Revised:

Accepted:

Published online:

^{1}; Stefan Lendl

^{2, 3}

@article{OJMO_2021__2__A3_0, author = {Marc Goerigk and Stefan Lendl}, title = {Robust {Combinatorial} {Optimization} with {Locally} {Budgeted} {Uncertainty}}, journal = {Open Journal of Mathematical Optimization}, eid = {3}, pages = {1--18}, publisher = {Universit\'e de Montpellier}, volume = {2}, year = {2021}, doi = {10.5802/ojmo.5}, language = {en}, url = {https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.5/} }

TY - JOUR AU - Marc Goerigk AU - Stefan Lendl TI - Robust Combinatorial Optimization with Locally Budgeted Uncertainty JO - Open Journal of Mathematical Optimization PY - 2021 SP - 1 EP - 18 VL - 2 PB - Université de Montpellier UR - https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.5/ DO - 10.5802/ojmo.5 LA - en ID - OJMO_2021__2__A3_0 ER -

%0 Journal Article %A Marc Goerigk %A Stefan Lendl %T Robust Combinatorial Optimization with Locally Budgeted Uncertainty %J Open Journal of Mathematical Optimization %D 2021 %P 1-18 %V 2 %I Université de Montpellier %U https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.5/ %R 10.5802/ojmo.5 %G en %F OJMO_2021__2__A3_0

Marc Goerigk; Stefan Lendl. Robust Combinatorial Optimization with Locally Budgeted Uncertainty. Open Journal of Mathematical Optimization, Volume 2 (2021), article no. 3, 18 p. doi : 10.5802/ojmo.5. https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.5/

[1] Min–max and min–max regret versions of combinatorial optimization problems: A survey, Eur. J. Oper. Res., Volume 197 (2009) no. 2, pp. 427-438 | DOI | MR | Zbl

[2] Some APX-completeness results for cubic graphs, Theor. Comput. Sci., Volume 237 (2000) no. 1-2, pp. 123-134 | DOI | MR | Zbl

[3] Robust constrained shortest path problems under budgeted uncertainty, Networks, Volume 66 (2015) no. 2, pp. 98-111 | DOI | MR | Zbl

[4] Data-driven robust optimization, Math. Program., Volume 167 (2018) no. 2, pp. 235-292 | DOI | MR | Zbl

[5] Robust discrete optimization and network flows, Math. Program., Volume 98 (2003) no. 1, pp. 49-71 | DOI | MR | Zbl

[6] The price of robustness, Oper. Res., Volume 52 (2004) no. 1, pp. 35-53 | DOI | MR | Zbl

[7] Robust scheduling with budgeted uncertainty, Discrete Appl. Math., Volume 261 (2019), pp. 93-107 | DOI | MR | Zbl

[8] New results about multi-band uncertainty in robust optimization, International Symposium on Experimental Algorithms (2012), pp. 63-74 | DOI | Zbl

[9] Algorithms and uncertainty sets for data-driven robust shortest path problems, Eur. J. Oper. Res., Volume 274 (2019) no. 2, pp. 671-686 | DOI | MR | Zbl

[10] On recoverable and two-stage robust selection problems with budgeted uncertainty, Eur. J. Oper. Res., Volume 265 (2018) no. 2, pp. 423-436 | DOI | MR | Zbl

[11] Faster algorithms for min-max-min robustness for combinatorial problems with budgeted uncertainty, Eur. J. Oper. Res., Volume 279 (2019) no. 2, pp. 308-319 | DOI | MR | Zbl

[12] Complexity and in-approximability of a selection problem in robust optimization, 4OR, Volume 11 (2013) no. 3, pp. 249-252 | DOI | MR | Zbl

[13] Approximating multiobjective knapsack problems, Manage. Sci., Volume 48 (2002) no. 12, pp. 1603-1612 | DOI | Zbl

[14] Computers and intractability, A Series of Books in the mathematical Sciences, 174, W. H. Freeman and Company, 1979 | Zbl

[15] Computational complexity of approximation algorithms for combinatorial problems, International Symposium on Mathematical Foundations of Computer Science (1979), pp. 292-300 | Zbl

[16] Recoverable Robust Representatives Selection Problems with Discrete Budgeted Uncertainty (2020) (https://arxiv.org/abs/2008.12727)

[17] An adaptive memory programming framework for the robust capacitated vehicle routing problem, Transportation Science, Volume 50 (2016) no. 4, pp. 1239-1260 | DOI

[18] Fast robust shortest path computations, 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018) (2018)

[19] Robust discrete optimization under discrete and interval uncertainty: A survey, Robustness analysis in decision aiding, optimization, and analytics, Springer, 2016, pp. 113-143 | DOI | Zbl

[20] Knapsack Problems, Springer, 2004 | Zbl

[21] Robust discrete optimization and its applications, 14, Springer, 2013 | Zbl

[22] Robust combinatorial optimization with variable budgeted uncertainty, 4OR, Volume 11 (2013) no. 1, pp. 75-92 | DOI | MR | Zbl

[23] Robust combinatorial optimization with knapsack uncertainty, Discrete Optimization, Volume 27 (2018), pp. 88-102 | DOI | MR | Zbl

[24] New and improved bounds for the minimum set cover problem, Approximation, randomization, and combinatorial optimization. Algorithms and techniques, Springer, 2012, pp. 288-300 | DOI | Zbl

[25] Exact algorithms for NP-hard problems: A survey, Combinatorial optimization – Eureka, you shrink! (Lecture Notes in Computer Science), Volume 2570, Springer, 2003, pp. 185-207 | DOI | MR | Zbl

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