We consider the following optimization problem: Given an instance of an optimization problem and some optimum solution for this instance, we want to find a good solution for a slightly modified instance. Additionally, the scenario is addressed where the solution for the original instance is not an arbitrary optimum solution, but is chosen among all optimum solutions in a most helpful way. In th...

We consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule. The machines have the same processing speed, but each job is compatible with only a subset of those machines. The machines can be linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can proce...

In the previous lecture, we saw how dynamic programming could be employed to obtain an FPTAS for the Knapsack problem. The key idea was to morph the given instance into another instance with additional structure — namely that the item profits weren’t too large — that allowed us to solve it exactly, and such that an optimal solution for the morphed instance could be used to construct a near-opti...

We study the approximability of the classical quadratic knapsack problem (QKP) on special graph classes. In this case the quadratic terms of the objective function are not given for each pair of knapsack items. Instead an edge weighted graph G = (V, E) whose vertices represent the knapsack items induces a quadratic profit p ij for the items i and j whenever they are adjacent in G (i.e (i, j) ∈ ...

Given a network with capacities and transit times on the arcs, the quickest flow problem asks for a “flow over time” that satisfies given demands within minimal time. In the setting of flows over time, flow on arcs may vary over time and the transit time of an arc is the time it takes for flow to travel through this arc. In most real-world applications (such as, e.g., road traffic, communicatio...

Operations research applications often pose multicriteria problems. Mathematical research on multicriteria problems predominantly revolves around the set of Pareto optimal solutions, while in practice, methods that output a single solution are more widespread. In real-world multicriteria optimization, reference point methods are widely used and successful examples of such methods. A reference p...

We extend the classical 0-1 knapsack problem by introducing disjunctive constraints for pairs of items which are not allowed to be packed together into the knapsack. These constraints are represented by edges of a conflict graph whose vertices correspond to the items of the knapsack problem. Similar conditions were treated in the literature for bin packing and scheduling problems. For the knaps...

We present an FPTAS for a combinatorial optimization problem which is motivated by a problem in drug-design. The problem is as follows. One is given two finite subsets A,B of points in three-dimensional space which represent the centers of atoms of two molecules. The objective function is large if there are an appropriate rigid motion M , subsets S ⊂ A and T ⊂ B of the same size and a bijective...

A flow on a directed network is said to be confluent if the flow uses at most one outgoing arc at each node. Confluent flows arise naturally from destination-based routing. We study the Maximum Confluent Flow Problem (MaxConf) with a single commodity but multiple sources and sinks. Unlike previous results, we consider heterogeneous arc capacities. The supplies and demands of the sources and sin...

Common approaches to solve a robust optimization problem decompose the problem into a master problem (MP) and adversarial separation problems (APs). MP contains the original robust constraints, however written only for finite numbers of scenarios. Additional scenarios are generated on the fly by solving the APs. We consider in this work the budgeted uncertainty polytope from Bertsimas and Sim, ...