The robust bilevel continuous knapsack problem with uncertain follower's objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower's problem. More precisely, adopting the robust optimization approach and assuming that the follower's profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower's reaction from the leader's perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader's objective function

The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower's Objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader's aim is to optimize a linear objective function in the capacity and in the follower's solution, but with respect to different item values. We address a stochastic version of this problem where the follower's profits are uncertain from the leader's perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input, which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader's objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items. Based on this, we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time.Comment: A preliminary version of parts of this article can be found in Section 8 of arXiv:1903.02810v

On the Complexity of the Bilevel Minimum Spanning Tree Problem

We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. By showing that this problem is NP-hard in general, we answer an open question stated by Shi et al. We prove that BMST remains hard even in the special case where the follower only controls a matching. Moreover, by a polynomial reduction from the vertex-disjoint Steiner trees problem, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a polynomial-time $(n-1)$-approximation algorithm for BMST, where $n$ is the number of vertices in the input graph. Moreover, considering the number of edges controlled by the follower as parameter, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting

On the complexity of the bilevel minimum spanning tree problem

We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. We show that this problem is NP-hard, even in the special case where the follower only controls a matching. Moreover, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a (|V|-1)-approximation algorithm for BMST, where |V| is the number of vertices. Moreover, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting

Faster Goal-Oriented Shortest Path Search for Bulk and Incremental Detailed Routing

We develop new algorithmic techniques for VLSI detailed routing. First, we improve the goal-oriented version of Dijkstra's algorithm to find shortest paths in huge incomplete grid graphs with edge costs depending on the direction and the layer, and possibly on rectangular regions. We devise estimates of the distance to the targets that offer better trade-offs between running time and quality than previously known methods, leading to an overall speed-up. Second, we combine the advantages of the two classical detailed routing approaches - global shortest path search and track assignment with local corrections - by treating input wires (such as the output of track assignment) as reservations that can be used at a discount by the respective net. We show how to implement this new approach efficiently

Partitioned vs. Integrated Planning of Hinterland Networks for LCL Transportation

Utilizing existing transportation networks better and designing (parts of) networks involves routing decisions to minimize transportation costs and maximize consolidation effects. We study the concrete example of hinterland networks for the truck-transportation of less-than-container-load (LCL) ocean freight shipments: A set of LCL shipments is given. They have to be routed through the hinterland network to be transported to an origin port and finally to the destination port via ship. On their way, they can be consolidated in hubs to full-container-load (FCL) shipments. The overall transportation cost depends on the selection of the origin port and the routing and consolidation in the hinterland network. A problem of this type appears for the global logistics provider DB Schenker. We translate the business problem into a hub location problem, describe it mathematically, and discuss solution strategies. As a result, an integrated modeling approach has several advantages over solving a simplified version of the problem, although it requires more computational effort

Controlled antibody release from gelatin for on-chip sample preparation

A practical way to realize on-chip sample preparation for point-of-care diagnostics is to store the required reagents on a microfluidic device and release them in a controlled manner upon contact with the sample. For the development of such diagnostic devices, a fundamental understanding of the release kinetics of reagents from suitable materials in microfluidic chips is therefore essential. Here, we study the release kinetics of fluorophore-conjugated antibodies from (sub-) µm thick gelatin layers and several ways to control the release time. The observed antibody release is well-described by a diffusion model. Release times ranging from ~20 s to ~650 s were determined for layers with thicknesses (in the dry state) between 0.25 µm and 1.5 µm, corresponding to a diffusivity of 0.65 µm2/s (in the swollen state) for our standard layer preparation conditions. By modifying the preparation conditions, we can influence the properties of gelatin to realize faster or slower release. Faster drying at increased temperatures leads to shorter release times, whereas slower drying at increased humidity yields slower release. As expected in a diffusive process, the release time increases with the size of the antibody. Moreover, the ionic strength of the release medium has a significant impact on the release kinetics. Applying these findings to cell counting chambers with on-chip sample preparation, we can tune the release to control the antibody distribution after inflow of blood in order to achieve homogeneous cell staining

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower’s Objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader’s aim is to optimize a linear objective function in the capacity and in the follower’s solution, but with respect to different item values. We address a stochastic version of this problem where the follower’s profits are uncertain from the leader’s perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input,which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader’s objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items.Based on this,we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time