58 research outputs found
The interval constrained 3-coloring problem
In this paper, we settle the open complexity status of interval constrained
coloring with a fixed number of colors. We prove that the problem is already
NP-complete if the number of different colors is 3. Previously, it has only
been known that it is NP-complete, if the number of colors is part of the input
and that the problem is solvable in polynomial time, if the number of colors is
at most 2. We also show that it is hard to satisfy almost all of the
constraints for a feasible instance.Comment: minor revisio
Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models
We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using to hide
polylogarithmic factors in (the number of nodes in the graph), our gradient
descent algorithm takes iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of . In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: -approximate SSSP using rounds, where is the (hop) diameter of the network.
(2) Broadcast congested clique model: -approximate
transshipment and SSSP using rounds. (3)
Multipass streaming model: -approximate transshipment and
SSSP using space and passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in ; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character
Providing Physical Layer Security for Mission Critical Machine Type Communication
The design of wireless systems for Mission Critical Machine Type
Communication (MC-MTC) is currently a hot research topic. Wireless systems are
considered to provide numerous advantages over wired systems in industrial
applications for example. However, due to the broadcast nature of the wireless
channel, such systems are prone to a wide range of cyber attacks. These range
from passive eavesdropping attacks to active attacks like data manipulation or
masquerade attacks. Therefore it is necessary to provide reliable and efficient
security mechanisms. One of the most important security issue in such a system
is to ensure integrity as well as authenticity of exchanged messages over the
air between communicating devices in order to prohibit active attacks. In the
present work, an approach on how to achieve this goal in MC-MTC systems based
on Physical Layer Security (PHYSEC), especially a new method based on keeping
track of channel variations, will be presented and a proof-of-concept
evaluation is given
Radio Link Enabler for Context-aware D2D Communication in Reuse Mode
Device-to-Device (D2D) communication is considered as one of the key
technologies for the fifth generation wireless communication system (5G) due to
certain benefits provided, e.g. traffic offload and low end-to-end latency. A
D2D link can reuse resource of a cellular user for its own transmission, while
mutual interference in between these two links is introduced. In this paper, we
propose a smart radio resource management (RRM) algorithm which enables D2D
communication to reuse cellular resource, by taking into account of context
information. Besides, signaling schemes with high efficiency are also given in
this work to enable the proposed RRM algorithm. Simulation results demonstrate
the performance improvement of the proposed scheme in terms of the overall cell
capacity
Two Results on Slime Mold Computations
We present two results on slime mold computations. In wet-lab experiments
(Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum
demonstrated its ability to solve shortest path problems. Biologists proposed a
mathematical model, a system of differential equations, for the slime's
adaption process (J. Theoretical Biology'07). It was shown that the process
convergences to the shortest path (J. Theoretical Biology'12) for all graphs.
We show that the dynamics actually converges for a much wider class of
problems, namely undirected linear programs with a non-negative cost vector.
Combinatorial optimization researchers took the dynamics describing slime
behavior as an inspiration for an optimization method and showed that its
discretization can -approximately solve linear programs with
positive cost vector (ITCS'16). Their analysis requires a feasible starting
point, a step size depending linearly on , and a number of steps
with quartic dependence on , where is
the difference between the smallest cost of a non-optimal basic feasible
solution and the optimal cost ().
We give a refined analysis showing that the dynamics initialized with any
strongly dominating point converges to the set of optimal solutions. Moreover,
we strengthen the convergence rate bounds and prove that the step size is
independent of , and the number of steps depends logarithmically
on and quadratically on
On Guillotine Cutting Sequences
Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack
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