Computing and minimizing the relative regret in combinatorial optimization with interval data

We consider combinatorial optimization problems with uncertain parameters of the objective function, where for each uncertain parameter an interval estimate is known. It is required to find a solution that minimizes the worst-case relative regret. For minmax relative regret versions of some subset-type problems, where feasible solutions are subsets of a finite ground set and the objective function represents the total weight of elements of a feasible solution, and for the minmax relative regret version of the problem of scheduling n jobs on a single machine to minimize the total completion time, we present a number of structural, algorithmic, and complexity results. Many of the results are based on generalizing and extending ideas and approaches from absolute regret minimization to the relative regret case.

[1]  Yijie Han,et al.  Concurrent threads and optimal parallel minimum spanning trees algorithm , 2001, JACM.

[2]  Tomasz Radzik.  Fractional Combinatorial Optimization , 1998 .

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

[4]  Nimrod Megiddo.  Combinatorial Optimization with Rational Objective Functions , 1979, Math. Oper. Res..

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

[6]  Eduardo Conde,et al.  An improved algorithm for selecting p items with uncertain returns according to the minmax-regret criterion , 2004, Math. Program..

[7]  Robert J. Vanderbei,et al.  Robust Optimization of Large-Scale Systems , 1995, Oper. Res..

[8]  Hande Yaman,et al.  The robust spanning tree problem with interval data , 2001, Oper. Res. Lett..

[9]  Igor Averbakh,et al.  Complexity of minimizing the total flow time with interval data and minmax regret criterion , 2006, Discret. Appl. Math..

[10]  Roberto Montemanni,et al.  A branch and bound algorithm for the robust shortest path problem with interval data , 2004, Oper. Res. Lett..

[11]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[12]  Pawel Zielinski,et al.  The computational complexity of the relative robust shortest path problem with interval data , 2004, Eur. J. Oper. Res..

[13]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[14]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[15]  Igor Averbakh,et al.  Interval data minmax regret network optimization problems , 2004, Discret. Appl. Math..

[16]  Manuel Laguna,et al.  Minimising the maximum relative regret for linear programmes with interval objective function coefficients , 1999, J. Oper. Res. Soc..

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

[18]  Roberto Montemanni,et al.  A branch and bound algorithm for the robust spanning tree problem with interval data , 2002, Eur. J. Oper. Res..

[19]  Igor Averbakh.  Minmax regret solutions for minimax optimization problems with uncertainty , 2000, Oper. Res. Lett..

[20]  Richard Cole,et al.  Slowing down sorting networks to obtain faster sorting algorithms , 2015, JACM.

[21]  Pascal Van Hentenryck,et al.  On the complexity of the robust spanning tree problem with interval data , 2004, Oper. Res. Lett..

[22]  Igor Averbakh.  The Minmax Relative Regret Median Problem on Networks , 2005, INFORMS J. Comput..

[23]  Panagiotis Kouvelis,et al.  Robust scheduling to hedge against processing time uncertainty in single-stage production , 1995 .

[24]  John H. Reif,et al.  Synthesis of Parallel Algorithms , 1993 .

[25]  Igor Averbakh,et al.  On the complexity of a class of combinatorial optimization problems with uncertainty , 2001, Math. Program..