Minimum Spanning Tree under Explorable Uncertainty in Theory and Experiments

We consider the minimum spanning tree (MST) problem in an uncertainty model where uncertain edge weights can be explored at extra cost. The task is to find an MST by querying a minimum number of edges for their exact weight. This problem has received quite some attention from the algorithms theory community. In this paper, we conduct the first practical experiments for MST under uncertainty, theoretically compare three known algorithms, and compare theoretical with practical behavior of the algorithms. Among others, we observe that the average performance and the absolute number of queries are both far from the theoretical worst-case bounds. Furthermore, we investigate a known general preprocessing procedure and develop an implementation thereof that maximally reduces the data uncertainty. We also characterize a class of instances that is solved completely by our preprocessing. Our experiments are based on practical data from an application in telecommunications and uncertainty instances generated from the standard TSPLib graph library

Set Selection under Explorable Stochastic Uncertainty via Covering Techniques

Given subsets of uncertain values, we study the problem of identifying the subset of minimum total value (sum of the uncertain values) by querying as few values as possible. This set selection problem falls into the field of explorable uncertainty and is of intrinsic importance therein as it implies strong adversarial lower bounds for a wide range of interesting combinatorial problems such as knapsack and matchings. We consider a stochastic problem variant and give algorithms that, in expectation, improve upon these adversarial lower bounds. The key to our results is to prove a strong structural connection to a seemingly unrelated covering problem with uncertainty in the constraints via a linear programming formulation. We exploit this connection to derive an algorithmic framework that can be used to solve both problems under uncertainty, obtaining nearly tight bounds on the competitive ratio. This is the first non-trivial stochastic result concerning the sum of unknown values without further structure known for the set. Further, we handle for the first time uncertainty in the constraints in a value-query model. With our novel methods, we lay the foundations for solving more general problems in the area of explorable uncertainty

Round-competitive algorithms for uncertainty problems with parallel queries

In computing with explorable uncertainty, one considers problems where the values of some input elements are uncertain, typically represented as intervals, but can be obtained using queries. Previous work has considered query minimization in the settings where queries are asked sequentially (adaptive model) or all at once (non-adaptive model). We introduce a new model where k queries can be made in parallel in each round, and the goal is to minimize the number of query rounds. Using competitive analysis, we present upper and lower bounds on the number of query rounds required by any algorithm in comparison with the optimal number of query rounds for the given instance. Given a set of uncertain elements and a family of m subsets of that set, we study the problems of sorting all m subsets and of determining the minimum value (or the minimum element(s)) of each subset. We also study the selection problem, i.e., the problem of determining the i-th smallest value and identifying all elements with that value in a given set of uncertain elements. Our results include 2-round-competitive algorithms for sorting and selection and an algorithm for the minimum value problem that uses at most (2 + ε) · optk + O 1 ε · lg m query rounds for every 0 < ε < 1, where optk is the optimal number of query round

Sorting and Hypergraph Orientation under Uncertainty with Predictions

Learning-augmented algorithms have been attracting increasing interest, but have only recently been considered in the setting of explorable uncertainty where precise values of uncertain input elements can be obtained by a query and the goal is to minimize the number of queries needed to solve a problem. We study learning-augmented algorithms for sorting and hypergraph orientation under uncertainty, assuming access to untrusted predictions for the uncertain values. Our algorithms provide improved performance guarantees for accurate predictions while maintaining worst-case guarantees that are best possible without predictions. For hypergraph orientation, for any $\gamma \geq 2$, we give an algorithm that achieves a competitive ratio of $1+1/\gamma$ for correct predictions and $\gamma$ for arbitrarily wrong predictions. For sorting, we achieve an optimal solution for accurate predictions while still being $2$-competitive for arbitrarily wrong predictions. These tradeoffs are the best possible. We also consider different error metrics and show that the performance of our algorithms degrades smoothly with the prediction error in all the cases where this is possible.Comment: arXiv admin note: text overlap with arXiv:2011.0738

Dynamics in Logistics

This open access book highlights the interdisciplinary aspects of logistics research. Featuring empirical, methodological, and practice-oriented articles, it addresses the modelling, planning, optimization and control of processes. Chiefly focusing on supply chains, logistics networks, production systems, and systems and facilities for material flows, the respective contributions combine research on classical supply chain management, digitalized business processes, production engineering, electrical engineering, computer science and mathematical optimization. To celebrate 25 years of interdisciplinary and collaborative research conducted at the Bremen Research Cluster for Dynamics in Logistics (LogDynamics), in this book hand-picked experts currently or formerly affiliated with the Cluster provide retrospectives, present cutting-edge research, and outline future research directions

Dynamics in Logistics

This open access book highlights the interdisciplinary aspects of logistics research. Featuring empirical, methodological, and practice-oriented articles, it addresses the modelling, planning, optimization and control of processes. Chiefly focusing on supply chains, logistics networks, production systems, and systems and facilities for material flows, the respective contributions combine research on classical supply chain management, digitalized business processes, production engineering, electrical engineering, computer science and mathematical optimization. To celebrate 25 years of interdisciplinary and collaborative research conducted at the Bremen Research Cluster for Dynamics in Logistics (LogDynamics), in this book hand-picked experts currently or formerly affiliated with the Cluster provide retrospectives, present cutting-edge research, and outline future research directions

Minimum Spanning Tree Verification under Uncertainty

In the verification under uncertainty setting, an algorithm is given, for each input item, an uncertainty area that is guaranteed to contain the exact input value, as well as an assumed input value. An update of an input item reveals its exact value. If the exact value is equal to the assumed value, we say that the update verifies the assumed value. We consider verification under uncertainty for the minimum spanning tree (MST) problem for undirected weighted graphs, where each edge is associated with an uncertainty area and an assumed edge weight. The objective of an algorithm is to compute the smallest set of updates with the property that, if the updates of all edges in the set verify their assumed weights, the edge set of an MST can be computed. We give a polynomial-time optimal algorithm for the MST verification problem by relating the choices of updates to vertex covers in a bipartite auxiliary graph. Furthermore, we consider an alternative uncertainty setting where the vertices are embedded in the plane, the weight of an edge is the Euclidean distance between the endpoints of the edge, and the uncertainty is about the location of the vertices. An update of a vertex yields the exact location of that vertex. We prove that the MST verification problem in this vertex uncertainty setting is NP-hard. This shows a surprising difference in complexity between the edge and vertex uncertainty settings of the MST verification problem