TY - CONF
AB - 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 \emph{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 \( \Omega ( \sqrt{|S|} + \frac{\log n}{\log \log n} ) \) that already holds for simple line metrics.
Here, \( n \) is the number of requests.
On the other side, we establish a deterministic \( \mathcal{O}(\sqrt{|S|} \cdot \log n) \)-competitive algorithm and a randomized \( \mathcal{O}(\sqrt{|S|} \cdot \frac{\log n}{\log \log n} ) \)-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.
AU - Castenow, Jannik
AU - Feldkord, Björn
AU - Knollmann, Till
AU - Malatyali, Manuel
AU - Meyer auf der Heide, Friedhelm
ID - 17370
KW - Online Multi-Commodity Facility Location
KW - Competitive Ratio
KW - Online Optimization
KW - Facility Location Problem
SN - 9781450369350
T2 - Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures
TI - The Online Multi-Commodity Facility Location Problem
ER -
TY - CONF
AU - Castenow, Jannik
AU - Kling, Peter
AU - Knollmann, Till
AU - Meyer auf der Heide, Friedhelm
ID - 17371
SN - 9781450369350
T2 - Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures
TI - Brief Announcement: A Discrete and Continuous Study of the Max-Chain-Formation Problem: Slow Down to Speed up
ER -
TY - CONF
AU - Pukrop, Simon
AU - Mäcker, Alexander
AU - Meyer auf der Heide, Friedhelm
ID - 13868
T2 - Proceedings of the 46th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM)
TI - Approximating Weighted Completion Time for Order Scheduling with Setup Times
ER -
TY - THES
AU - Feldkord, Björn
ID - 15631
TI - Mobile Resource Allocation
ER -
TY - CONF
AB - 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.
AU - Baswana, Surender
AU - Gupta, Shiv
AU - Knollmann, Till
ED - Grandoni, Fabrizio
ED - Herman, Grzegorz
ED - Sanders, Peter
ID - 20159
KW - Mincut
KW - Sensitivity
KW - Data Structure
SN - 1868-8969
T2 - 28th Annual European Symposium on Algorithms (ESA 2020)
TI - Mincut Sensitivity Data Structures for the Insertion of an Edge
VL - 173
ER -
TY - CONF
AB - 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 $\Theta(n^2)$ 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
$O(\Delta \cdot n^{3/2})$ for any three-dimensional contracting strategy, where
$\Delta$ denotes the diameter of the initial configuration. This comes up to a
factor of $\sqrt{n}$ close to the lower bound of $\Omega (\Delta \cdot n)$
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.
AU - Braun, Michael
AU - Castenow, Jannik
AU - Meyer auf der Heide, Friedhelm
ID - 16968
T2 - Proceedings of the 27th Conference on Structural Information and Communication Complexity (SIROCCO)
TI - Local Gathering of Mobile Robots in Three Dimensions
ER -
TY - CONF
AU - Castenow, Jannik
AU - Harbig, Jonas
AU - Jung, Daniel
AU - Knollmann, Till
AU - Meyer auf der Heide, Friedhelm
ID - 20185
T2 - Stabilization, Safety, and Security of Distributed Systems - 21st International Symposium, SSS 2020, Austin, Texas, USA, November 18-21, 2020, Proceedings (accepted)
TI - Brief Announcement: Gathering in Linear Time: A Closed Chain of Disoriented & Luminous Robots with Limited Visibility
ER -
TY - CONF
AU - Castenow, Jannik
AU - Kolb, Christina
AU - Scheideler, Christian
ID - 15169
T2 - Proceedings of the 21st International Conference on Distributed Computing and Networking (ICDCN)
TI - A Bounding Box Overlay for Competitive Routing in Hybrid Communication Networks
ER -
TY - JOUR
AU - Castenow, Jannik
AU - Fischer, Matthias
AU - Harbig, Jonas
AU - Jung, Daniel
AU - Meyer auf der Heide, Friedhelm
ID - 16299
JF - Theoretical Computer Science
SN - 0304-3975
TI - Gathering Anonymous, Oblivious Robots on a Grid
VL - 815
ER -
TY - CONF
AB - 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.
AU - Castenow, Jannik
AU - Kling, Peter
AU - Knollmann, Till
AU - Meyer auf der Heide, Friedhelm
ID - 19899
T2 - Stabilization, Safety, and Security of Distributed Systems - 21st International Symposium, SSS 2020, Austin, Texas, USA, November 18-21, 2020, Proceedings (accepted)
TI - A Discrete and Continuous Study of the Max-Chain-Formation Problem
ER -