Robust multicovers with budgeted uncertainty

Abstract The Min-q-Multiset Multicover problem presented in this paper is a special version of the Multiset Multicover problem. For a fixed positive integer q, we are given a finite ground set J, an integral demand for each element in J and a collection of subsets of J. The task is to choose sets of the collection (multiple choices are allowed) such that each element in J is covered at least as many times as specified by the demand of the element. In contrast to Multiset Multicover, in Min-q-Multiset Multicover each of the chosen subsets may only cover up to q of its elements with multiple choices being allowed. Our main focus is a robust version of Min-q-Multiset Multicover, called Robust Min-q-Multiset Multicover, in which the demand of each element in J may vary in a given interval with an additional budget constraint bounding the sum of the demands. Again, the task is to find a selection of subsets which is feasible for all admissible demands. We show that the non-robust version is NP-complete for q greater than two, whereas the robust version is strongly NP-hard for any positive q. Furthermore, we present two solution approaches based on constraint generation and investigate the corresponding separation problems. We present computational results using randomly generated instances as well as instances emerging from the problem of locating emergency doctors.

[1]  Laurent El Ghaoui,et al.  Robust Optimization , 2021, ICORES.

[2]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

[3]  R. Ravi,et al.  Thresholded Covering Algorithms for Robust and Max-min Optimization , 2010, ICALP.

[4]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[5]  Matteo Fischetti,et al.  Cutting plane versus compact formulations for uncertain (integer) linear programs , 2012, Math. Program. Comput..

[6]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[7]  Laurence A. Wolsey,et al.  Production Planning by Mixed Integer Programming (Springer Series in Operations Research and Financial Engineering) , 2006 .

[8]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[9]  Arkadi Nemirovski,et al.  Robust solutions of uncertain linear programs , 1999, Oper. Res. Lett..

[10]  Paweł Zieliński,et al.  Robust Discrete Optimization Under Discrete and Interval Uncertainty: A Survey , 2016 .

[11]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[12]  Ravindra K. Ahuja,et al.  Network Flows , 2011 .

[13]  Adam Kasperski,et al.  On Recoverable and Two-Stage Robust Selection Problems with Budgeted Uncertainty , 2018, Eur. J. Oper. Res..

[14]  Allen L. Soyster,et al.  Technical Note - Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming , 1973, Oper. Res..

[15]  Joseph Naor,et al.  Online Primal-Dual Algorithms for Covering and Packing Problems , 2005, ESA.

[16]  Brigitte Werners,et al.  Improved handling of uncertainty and robustness in set covering problems , 2017, Eur. J. Oper. Res..

[17]  Laurence A. Wolsey,et al.  Production Planning by Mixed Integer Programming , 2010 .

[18]  Francis C. M. Lau,et al.  Set multi-covering via inclusion-exclusion , 2009, Theor. Comput. Sci..

[19]  Cécile Murat,et al.  Recent advances in robust optimization: An overview , 2014, Eur. J. Oper. Res..

[20]  Vahab S. Mirrokni,et al.  Robust Combinatorial Optimization with Exponential Scenarios , 2007, IPCO.

[21]  Arkadi Nemirovski,et al.  Robust solutions of Linear Programming problems contaminated with uncertain data , 2000, Math. Program..

[22]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[23]  Vijay V. Vazirani,et al.  Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs , 1999, SIAM J. Comput..

[24]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[25]  A. Ben-Tal,et al.  Adjustable robust solutions of uncertain linear programs , 2004, Math. Program..

[26]  Michel Minoux,et al.  On 2-stage robust LP with RHS uncertainty: complexity results and applications , 2011, J. Glob. Optim..

[27]  Michael Poss,et al.  Robust scheduling with budgeted uncertainty , 2019, Discret. Appl. Math..

[28]  Melvyn Sim,et al.  Robust discrete optimization and network flows , 2003, Math. Program..

[29]  Cécile Murat,et al.  Robust location transportation problems under uncertain demands , 2014, Discret. Appl. Math..

[30]  Melvyn Sim,et al.  The Price of Robustness , 2004, Oper. Res..

[31]  Francis C. M. Lau,et al.  Exact Algorithms for Set Multicover and Multiset Multicover Problems , 2009, ISAAC.

[32]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[33]  Adam Kasperski,et al.  Discrete Optimization with Interval Data - Minmax Regret and Fuzzy Approach , 2008, Studies in Fuzziness and Soft Computing.

[34]  A Gerodimos,et al.  Robust Discrete Optimization and its Applications , 1996, J. Oper. Res. Soc..

[35]  Ebrahim Nasrabadi,et al.  Robust optimization with incremental recourse , 2013, ArXiv.

[36]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[37]  Noga Alon,et al.  THE ONLINE SET COVER PROBLEM∗ , 2009 .

[38]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[39]  Andrew V. Goldberg,et al.  Maximum Skew-Symmetric Flows , 1995, ESA.