Efficient algorithms for robustness in resource allocation and scheduling problems

AbstractThe robustness function of an optimization (minimization) problem measures the maximum increase in the value of its optimal solution that can be produced by spending a given amount of resources increasing the values of the elements in its input. We present efficient algorithms for computing the robustness function of resource allocation and scheduling problems that can be modeled with partition and scheduling matroids. For the case of scheduling matroids, we give an O(m2n2) time algorithm for computing a complete description of the robustness function, where m is the number of elements in the matroid and n is its rank. For partition matroids, we give two algorithms: one that computes the complete robustness function in O(mlogm) time, and other that optimally evaluates the robustness function at only a specified point

Bottleneck Capacity Expansion Problems with General Budget Constraints

This paper presents a unified approach for bottleneck capacity expansion problems. In the bottleneck capacity expansion problem, BCEP, we are given a finite ground set E, a family F of feasible subsets of E and a nonnegative real capacity ĉe for all e ∈ E. Moreover, we are given monotone increasing cost functions fe for increasing the capacity of the elements e ∈ E as well as a budget B. The task is to determine new capacities ce ≥ ĉe such that the objective function given by maxF∈Fmine∈Fce is maximized under the side constraint that the overall expansion cost does not exceed the budget B. We introduce an algebraic model for defining the overall expansion cost and for formulating the budget constraint. This models allows to capture various types of budget constraints in one general model. Moreover, we discuss solution approaches for the general bottleneck capacity expansion problem. For an important subclass of bottleneck capacity expansion problems we propose algorithms which perform a strongly polynomial number of steps. In this manner we generalize and improve a recent result of Zhang et al. [15]

Bottleneck Capacity Expansion Problems With General Budget Constraints

This paper presents a unified approach for bottleneck capacity expansion problems. In the bottleneck capacity expansion problem, BCEP, we are given a finite ground set E, a family F of feasible subsets of E and a nonnegative real capacity b c e for all e 2 E. Moreover, we are given monotone increasing cost functions f e for increasing the capacity of the elements e 2 E as well as a budget B. The task is to determine new capacities c e b c e such that the objective function given by max F2F min e2F c e is maximized under the side constraint that the overall expansion cost does not exceed the budget B. We introduce an algebraic model for defining the overall expansion cost and for formulating the budget constraint. This models allows to capture various types of budget constraints in one general model. Moreover, we discuss solution approaches for the general bottleneck capacity expansion problem. For an important subclass of bottleneck capacity expansion problems we propose ..