Abstract A classical result by Myerson (Math. Oper. Res. 6(1), 58-73, 1981) gives a characterization of an optimal auction for any given distribution of valuations of the bidders. We consider the situation where the distribution is not explicitly given but can be observed in a sample of auction results from the same distribution. A seminal paper by Morgenstern and Roughgarden (Adv.Neural Inf. Process. Syst. 28, 2015) proposes to learn a near-optimal auction from the hypothesis class of t-level auctions. They prove a bound on the sample complexity, i.e., the function $$f(\varepsilon , \delta )$$ f ( ε , δ ) of required samples to guarantee a certain level of precision $$(1-\varepsilon )$$ ( 1 - ε ) with a probability of at least $$(1-\delta )$$ ( 1 - δ ) , for the general single-parameter case and a tighter bound for the very restricted matroid case. We show a new bound for the case of independence systems, that widely generalizes matroids and contains several important combinatorial optimization problems. This bound of $$\tilde{O}\left( \nicefrac {H^2n^4}{\varepsilon ^3}\right) $$ O ~ H 2 n 4 ε 3 falls neatly between those known for the general and the matroid case. The class of independence systems contains several well known NP-hard problems such as knapsack. Therefore, the allocation itself might in practice be limited to $$\alpha $$ α -approximate solutions. In a second result we show that an approximation algorithm can be used without compromising the sample complexity. Also, the precision is affected only mildly, resulting in a factor of $$\alpha \cdot (1-\varepsilon )$$ α · ( 1 - ε ) .

We present an efficient and effective computational frame-work for the inverse rendering problem of reconstructing the 3D shape of a piece of glass from its caustic image. Our approach is motivated by the needs of 3D glass printing, a nascent additive manufacturing technique that promises to revolutionize the production of optics elements, from lightweight mirrors to waveguides and lenses. One important problem is the reliable control of the manufacturing process by inferring the printed 3D glass shape from its caustic image. Towards this goal, we propose a novel general-purpose reconstruction algorithm based on differentiable light propagation simulation followed by a regularization scheme that takes the deposited glass volume into account. This enables incorporating arbitrary measurements of caustics into an efficient reconstruction framework. We demonstrate the effectiveness of our method and establish the influence of our hyperparameters using several sample shapes and parameter configurations.

We develop a fast method to compute an optimal robust shortest path in large networks like road networks, a fundamental problem in traffic and logistics under uncertainty. In the robust shortest path problem we are given an s-t-graph D(V,A) and for each arc a nominal length c(a) and a maximal increase d(a) of its length. We consider all scenarios in which for the increased lengths c(a) + bar{d}(a) we have bar{d}(a) <= d(a) and sum_{a in A} (bar{d}(a)/d(a)) <= Gamma. Each path is measured by the length in its worst-case scenario. A classic result [Bertsimas and Sim, 2003] minimizes this path length by solving (|A| + 1)-many shortest path problems. Easily, (|A| + 1) can be replaced by |Theta|, where Theta is the set of all different values d(a) and 0. Still, the approach remains impractical for large graphs. Using the monotonicity of a part of the objective we devise a Divide and Conquer method to evaluate significantly fewer values of Theta. This methods generalizes to binary linear robust problems. Specifically for shortest paths we derive a lower bound to speed-up the Divide and Conquer of Theta. The bound is based on carefully using previous shortest path computations. We combine the approach with non-preprocessing based acceleration techniques for Dijkstra adapted to the robust case. In a computational study we document the value of different accelerations tried in the algorithm engineering process. We also give an approximation scheme for the robust shortest path problem which computes a (1 + epsilon)-approximate solution requiring O(log(d^ / (1 + epsilon))) computations of the nominal problem where d^ := max d(A) / min (d(A)\{0}).

We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio $\varphi\approx1.618$. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items in the beginning and try to pack the items in this order, without changing the order later on. We give efficient algorithms computing these policies. On the other hand, we show that, for any $\alpha>1$, the problem of deciding whether a given universal policy achieves a factor of $\alpha$ is ${\mathsf{coNP}}$-complete. If $\alpha$ is part of the input, the same problem is shown to be ${\mathsf{coNP}}$-complete for items with unit densities. Finally, we show that it is ${\mathsf{coNP}}$-hard to decide, for given $\alpha$, whether a set of items admits a universal policy with factor $\alpha$, even if all items have unit densities.

Health care providers are under tremendous pressure to reduce costs and increase quality of their services. It has long been recognized that well-designed appointment systems have the potential to improve utilization of expensive personnel and medical equipment and to reduce waiting times for patients. In a widely influential survey on outpatient scheduling, Cayirli and Veral (2003) concluded that the "biggest challenge for future research will be to develop easy-to-use heuristics." We analyze the appointment scheduling problem from a robust-optimization perspective, and we establish the existence of a closed-form optimal solution--arguably the simplest and best `heuristic' possible. In case the order of patients is changeable, the robust optimization approach yields a novel formulation of the appointment scheduling problem as that of minimizing a concave function over a supermodular polyhedron. We devise the first constant-factor approximation algorithm for this case.

We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items and try to pack the items in this order, independent of the observations made while packing. We give efficient algorithms computing these policies. On the other hand, we show that, for any a>1, the problem of deciding whether a given universal policy achieves a factor of a is coNP-complete. If a is part of the input, the same problem is shown to be coNP-complete for items with unit densities. Finally, we show that it is coNP-hard to decide, for given a, whether a set of items admits a universal policy with factor a, even if all items have unit densities.

Real-time systems increasingly contain processing units with multiple cores. To use this additional computational power in hard deadline environments, one needs schedulability tests for task models that represent the possibilities of parallel execution of jobs of a task. A standard model is to represent a (sporadically) recurrent task by a directed a cyclic graph (DAG). The nodes of the DAG correspond to the jobs of the task. All such jobs are released simultaneously, have to be completed within some common relative deadline, and some pairs of jobs are linked by a precedence constraint, i.e., an arc of the DAG. This poses new challenges for analyzing whether a task system is feasible, in particular for the commonly used online algorithms Earliest Deadline First (EDF) and Deadline Monotonic (DM). While for ordinary sporadic tasks the required algorithmic techniques are well-understood, despite recent research much remains open in this model. In this work, we completely close the gap between the algorithmic understanding of feasibility analysis for the usual sporadic task model and the case where each sporadic task is a DAG. We show for DAG tasks that EDF has a tight speedup bound of 2 - 1/m, where m is the number of processors, while DM has a speedup bound of at most 3 - 1/m. Moreover, we present polynomial and pseudopolynomial time tests, of differing effectiveness, for determining whether a set of sporadic DAG tasks can be scheduled by EDF or DM to meet all deadlines on a specified number of processors. We remark that the effectiveness of some of our tests matches the best known algorithms for ordinary sporadic task sets, thus closing the gap.

Abstract We consider a basic subproblem which arises in line planning, and is of particular importance in the context of a high system load or robustness: How much can be routed maximally along all possible lines? The essence of this problem is the path constrained network flow (PCN) problem. We explore the complexity of this problem and its dual. In particular we show for the primal that it is as hard to approximate as MAX CLIQUE and for the dual that it is as hard to approximate as SET COVER. We also prove that the PCN problem is hard for special graph classes, interesting both from a complexity and from a practical perspective. Finally, we present a special graph class for which there is a polynomial‐time algorithm. © 2010 Wiley Periodicals, Inc. NETWORKS, 2011

Recent results have demonstrated the existence of a sufficient global EDF schedulability test for sporadic task systems that makes the following guarantee: any task system that is not determined to be schedulable on an m-processor platform by this test is guaranteed to actually not be so on a platform in which each processor is m/(2m - 1) times as fast. A new global EDF schedulability test is proposed here that builds on this result. This new test is shown to be less pessimistic and more widely applicable than the earlier result was, while retaining the strong theoretical properties - in particular, the speedup bound - of the earlier result.