A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location - Bienkowski, Marcin; Feldkord, Björn; Schmidt, Pawel (Proceedings of the 38th Symposium on Theoretical Aspects of Computer Science (STACS))

A continuous strategy for collisionless gathering - Li, Shouwei; Markarian, Christine; Meyer auf der Heide, Friedhelm; Podlipyan, Pavel (Theoretical Computer Science)
Over the past decades, the Gathering problem, which asks to gather a group of robots in finite time given some restrictions, has been intensively studied. In this paper, we are given a group of n autonomous, dimensionless, deterministic, and anonymous robots, with bounded viewing range. Assuming a continuous time model, the goal is to gather these robots into one point in finite time. We introduce a simple convergence criterion that defines a new class of algorithms which perform gathering in O(nd) time, where d is the diameter of the initial robot configuration. We show that some gathering algorithms in the literature belong to this class and propose two new algorithms that belong to this class and have quadratic running time, namely, Go-To-The-Relative-Center algorithm (GTRC) and Safe-Go-To-The-Relative-Center algorithm (S-GTRC). We prove that the latter can perform gathering without collision by using a slightly more complex robot model: non oblivious, chiral, and luminous (i.e. robots have observable external memory, as in [8]). We also consider a variant of the Gathering problem, the Near-Gathering problem, in which robots must get close to each other without colliding. We show that S-GTRC solves the Near-Gathering problem in quadratic time and assumes a weaker robot model than the one assumed in the current state-of-the-art.
Keywords: Local algorithms; Distributed algorithms; Collisionless gathering; Mobile robots; Multiagent system

The impact of the Gabriel subgraph of the visibility graph on the gathering of mobile autonomous robots - Li, Shouwei; Meyer auf der Heide, Friedhelm; Podlipyan, Pavel (Theoretical Computer Science)
In this paper, we reconsider the well-known discrete, round-based Go-To-The-Center algorithm due to Ando, Suzuki, and Yamashita [2] for gathering n autonomous mobile robots with limited viewing range in the plane. Remarquably, this algorithm exploits the fact that during its execution, many collisions of robots occur. Such collisions are interpreted as a success because it is assumed that such collided robots behave the same from now on. This is acceptable under the assumption that each robot is represented by a single point. Otherwise, collisions should be avoided. In this paper, we consider a continuous Go-To-The-Center algorithm in which the robots continuously observe the positions of their neighbors and adapt their speed (assuming a speed limit) and direction. Our first results are time bounds of O(n2) for gathering in two dimensions Euclidean space, and Θ(n) for the one dimension. Our main contribution is the introduction and evaluation of a continuous algorithm which performs Go-To-The-Center considering only the neighbors of a robot with respect to the Gabriel subgraph of the visibility graph, i.e. Go-To-The-Gabriel-Center algorithm. We show that this modification still correctly executes gathering in one and two dimensions, with the same time bounds as above. Simulations exhibit a severe difference of the behavior of the Go-To-The-Center and the Go-To-The-Gabriel-Center algorithms: Whereas lots of collisions occur during a run of the Go-To-The-Center algorithm, typically only one, namely the final collision occurs during a run of the Go-To-The-Gabriel-Center algorithm. We can prove this “collisionless property” of the Go-To-The-Gabriel-Center algorithm for one dimension. In two-dimensional Euclidean space, we conjecture that the “collisionless property” holds for almost every initial configuration. We support our conjecture with measurements obtained from the simulation where robots execute both continuous Go-To-The-Center and Go-To-The-Gabriel-Center algorithms.
Keywords: Local algorithms; Distributed algorithms; Collisionless gathering; Mobile robots; Multiagent system

The Max-Line-Formation Problem – And New Insights for Gathering and Chain-Formation - Castenow, Jannik; Götte, Thorsten; Knollmann, Till; Meyer auf der Heide, Friedhelm (Proceedings of the 23rd International Symposium on Stabilization, Safety, and Security of Distributed Systems, SSS 2021)

Full Version -- Server Cloud Scheduling - Maack, Marten; Meyer auf der Heide, Friedhelm; Pukrop, Simon (arXiv:2108.02109)
Consider a set of jobs connected to a directed acyclic task graph with a fixed source and sink. The edges of this graph model precedence constraints and the jobs have to be scheduled with respect to those. We introduce the Server Cloud Scheduling problem, in which the jobs have to be processed either on a single local machine or on one of many cloud machines. Both the source and the sink have to be scheduled on the local machine. For each job, processing times both on the server and in the cloud are given. Furthermore, for each edge in the task graph, a communication delay is included in the input and has to be taken into account if one of the two jobs is scheduled on the server, the other in the cloud. The server can process jobs sequentially, whereas the cloud can serve as many as needed in parallel, but induces costs. We consider both makespan and cost minimization. The main results are an FPTAS with respect for the makespan objective for a fairly general case and strong hardness for the case with unit processing times and delays.

Combining Mobility, Heterogeneity, and Leasing Approaches for Online Resource Allocation - Berger, Thilo Frederik

Online k-Facility Reallocation using k-Server Algorithms - Pranger, Sebastian

A Discrete and Continuous Study of the Max-Chain-Formation Problem – Slow Down to Speed Up - Castenow, Jannik; Kling, Peter; Knollmann, Till; Meyer auf der Heide, Friedhelm (Stabilization, Safety, and Security of Distributed Systems - 22nd International Symposium, SSS 2020, Austin, Texas, USA, November 18-21, 2020, Proceedings)
Most existing robot formation problems seek a target formation of a certain minimal and, thus, efficient structure. Examples include the Gathering and the Chain-Formation problem. In this work, we study formation problems that try to reach a maximal structure, supporting for example an efficient coverage in exploration scenarios. A recent example is the NASA Shapeshifter project, which describes how the robots form a relay chain along which gathered data from extraterrestrial cave explorations may be sent to a home base. As a first step towards understanding such maximization tasks, we introduce and study the Max-Chain-Formation problem, where $n$ robots are ordered along a winding, potentially self-intersecting chain and must form a connected, straight line of maximal length connecting its two endpoints. We propose and analyze strategies in a discrete and in a continuous time model. In the discrete case, we give a complete analysis if all robots are initially collinear, showing that the worst-case time to reach an $\varepsilon$-approximation is upper bounded by $\mathcal{O}(n^2 \cdot \log (n/\varepsilon))$ and lower bounded by $\Omega(n^2 \cdot~\log (1/\varepsilon))$. If one endpoint of the chain remains stationary, this result can be extended to the non-collinear case. If both endpoints move, we identify a family of instances whose runtime is unbounded. For the continuous model, we give a strategy with an optimal runtime bound of $\Theta(n)$. Avoiding an unbounded runtime similar to the discrete case relies crucially on a counter-intuitive aspect of the strategy: slowing down the endpoints while all other robots move at full speed. Surprisingly, we can show that a similar trick does not work in the discrete model.

Mincut Sensitivity Data Structures for the Insertion of an Edge - Baswana, Surender; Gupta, Shiv; Knollmann, Till (28th Annual European Symposium on Algorithms (ESA 2020))
Let G = (V,E) be an undirected graph on n vertices with non-negative capacities on its edges. The mincut sensitivity problem for the insertion of an edge is defined as follows. Build a compact data structure for G and a given set S ⊆ V of vertices that, on receiving any edge (x,y) ∈ S×S of positive capacity as query input, can efficiently report the set of all pairs from S× S whose mincut value increases upon insertion of the edge (x,y) to G. The only result that exists for this problem is for a single pair of vertices (Picard and Queyranne, Mathematical Programming Study, 13 (1980), 8-16). We present the following results for the single source and the all-pairs versions of this problem. 1) Single source: Given any designated source vertex s, there exists a data structure of size 𝒪(|S|) that can output all those vertices from S whose mincut value to s increases upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(|S|). 2) All-pairs: There exists an 𝒪(|S|²) size data structure that can output all those pairs of vertices from S× S whose mincut value gets increased upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(k), where k is the number of pairs of vertices whose mincut increases. For both these versions, we also address the problem of reporting the values of the mincuts upon insertion of any given edge. To derive our results, we use interesting insights into the nearest and the farthest mincuts for a pair of vertices. In addition, a crucial result, that we establish and use in our data structures, is that there exists a directed acyclic graph of 𝒪(n) size that compactly stores the farthest mincuts from all vertices of V to a designated vertex s in the graph. We believe that this result is of independent interest, especially, because it also complements a previously existing result by Hariharan et al. (STOC 2007) that the nearest mincuts from all vertices of V to s is a laminar family, and hence, can be stored compactly in a tree of 𝒪(n) size.
Keywords: Mincut; Sensitivity; Data Structure

Brief Announcement: Gathering in Linear Time: A Closed Chain of Disoriented & Luminous Robots with Limited Visibility - Castenow, Jannik; Harbig, Jonas; Jung, Daniel; Knollmann, Till; Meyer auf der Heide, Friedhelm (Stabilization, Safety, and Security of Distributed Systems - 22nd International Symposium, SSS 2020, Austin, Texas, USA, November 18-21, 2020, Proceedings)

The Online Multi-Commodity Facility Location Problem - Castenow, Jannik; Feldkord, Björn; Knollmann, Till; Malatyali, Manuel; Meyer auf der Heide, Friedhelm (Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures)
We consider a natural extension to the metric uncapacitated Facility Location Problem (FLP) in which requests ask for different commodities out of a finite set \( S \) of commodities. Ravi and Sinha (SODA 2004) introduced the model as the Multi-Commodity Facility Location Problem (MFLP) and considered it an offline optimization problem. The model itself is similar to the FLP: i.e., requests are located at points of a finite metric space and the task of an algorithm is to construct facilities and assign requests to facilities while minimizing the construction cost and the sum over all assignment distances. In addition, requests and facilities are heterogeneous; they request or offer multiple commodities out of $S$. A request has to be connected to a set of facilities jointly offering the commodities demanded by it. In comparison to the FLP, an algorithm has to decide not only if and where to place facilities, but also which commodities to offer at each. To the best of our knowledge we are the first to study the problem in its online variant in which requests, their positions and their commodities are not known beforehand but revealed over time. We present results regarding the competitive ratio. On the one hand, we show that heterogeneity influences the competitive ratio by developing a lower bound on the competitive ratio for any randomized online algorithm of \( Ω ( \sqrt{|S|} + \frac{\log n}{\log \log n} ) \) that already holds for simple line metrics. Here, \( n \) is the number of requests. On the other side, we establish a deterministic \( \mathcal{O}(\sqrt{|S|} · \log n) \)-competitive algorithm and a randomized \( \mathcal{O}(\sqrt{|S|} · \frac{\log n}{\log \log n} ) \)-competitive algorithm. Further, we show that when considering a more special class of cost functions for the construction cost of a facility, the competitive ratio decreases given by our deterministic algorithm depending on the function.
Keywords: Online Multi-Commodity Facility Location; Competitive Ratio; Online Optimization; Facility Location Problem

Brief Announcement: A Discrete and Continuous Study of the Max-Chain-Formation Problem: Slow Down to Speed up - Castenow, Jannik; Kling, Peter; Knollmann, Till; Meyer auf der Heide, Friedhelm (Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures)

Local Gathering of Mobile Robots in Three Dimensions - Braun, Michael; Castenow, Jannik; Meyer auf der Heide, Friedhelm (Proceedings of the 27th Conference on Structural Information and Communication Complexity (SIROCCO))
In this work, we initiate the research about the Gathering problem for robots with limited viewing range in the three-dimensional Euclidean space. In the Gathering problem, a set of initially scattered robots is required to gather at the same position. The robots' capabilities are very restricted -- they do not agree on any coordinate system or compass, have a limited viewing range, have no memory of the past and cannot communicate. We study the problem in two different time models, in FSYNC (fully synchronized discrete rounds) and the continuous time model. For FSYNC, we introduce the 3D-Go-To-The-Center-strategy and prove a runtime of $Θ(n^2)$ that matches the currently best runtime bound for the same model in the Euclidean plane [SPAA'11]. Our main result is the generalization of contracting strategies (continuous time) from [Algosensors'17] to three dimensions. In contracting strategies, every robot that is located on the global convex hull of all robots' positions moves with full speed towards the inside of the convex hull. We prove a runtime bound of $O(Δ · n^{3/2})$ for any three-dimensional contracting strategy, where $Δ$ denotes the diameter of the initial configuration. This comes up to a factor of $\sqrt{n}$ close to the lower bound of $Ω (Δ · n)$ which is already true in two dimensions. In general, it might be hard for robots with limited viewing range to decide whether they are located on the global convex hull and which movement maintains the connectivity of the swarm, rendering the design of concrete contracting strategies a challenging task. We prove that the continuous variant of 3D-Go-To-The-Center is contracting and keeps the swarm connected. Moreover, we give a simple design criterion for three-dimensional contracting strategies that maintains the connectivity of the swarm and introduce an exemplary strategy based on this criterion.

Mobile Resource Allocation - Feldkord, Björn

Multi-Oracle Coevolutionary Learning of Requirements Specifications from Examples in On-The-Fly Markets - Wever, Marcel Dominik; van Rooijen, Lorijn; Hamann, Heiko (Evolutionary Computation)
In software engineering, the imprecise requirements of a user are transformed to a formal requirements specification during the requirements elicitation process. This process is usually guided by requirements engineers interviewing the user. We want to partially automate this first step of the software engineering process in order to enable users to specify a desired software system on their own. With our approach, users are only asked to provide exemplary behavioral descriptions. The problem of synthesizing a requirements specification from examples can partially be reduced to the problem of grammatical inference, to which we apply an active coevolutionary learning approach. However, this approach would usually require many feedback queries to be sent to the user. In this work, we extend and generalize our active learning approach to receive knowledge from multiple oracles, also known as proactive learning. The ‘user oracle’ represents input received from the user and the ‘knowledge oracle’ represents available, formalized domain knowledge. We call our two-oracle approach the ‘first apply knowledge then query’ (FAKT/Q) algorithm. We compare FAKT/Q to the active learning approach and provide an extensive benchmark evaluation. As result we find that the number of required user queries is reduced and the inference process is sped up significantly. Finally, with so-called On-The-Fly Markets, we present a motivation and an application of our approach where such knowledge is available.

A Bounding Box Overlay for Competitive Routing in Hybrid Communication Networks - Castenow, Jannik; Kolb, Christina; Scheideler, Christian (Proceedings of the 21st International Conference on Distributed Computing and Networking (ICDCN))

Gathering Anonymous, Oblivious Robots on a Grid - Castenow, Jannik; Fischer, Matthias; Harbig, Jonas; Jung, Daniel; Meyer auf der Heide, Friedhelm (Theoretical Computer Science)

Approximating Weighted Completion Time for Order Scheduling with Setup Times - Pukrop, Simon; Mäcker, Alexander; Meyer auf der Heide, Friedhelm (Proceedings of the 46th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM))

A Case for a New IT Ecosystem: On-The-Fly Computing - Karl, Holger; Kundisch, Dennis; Meyer auf der Heide, Friedhelm; Wehrheim, Heike (Business & Information Systems Engineering)

Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient - Baswana, Surender; Gupta, Shiv; Tulsyan, Ayush (44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019))