2,498 research outputs found
Complexity of the General Chromatic Art Gallery Problem
In the original Art Gallery Problem (AGP), one seeks the minimum number of
guards required to cover a polygon . We consider the Chromatic AGP (CAGP),
where the guards are colored. As long as is completely covered, the number
of guards does not matter, but guards with overlapping visibility regions must
have different colors. This problem has applications in landmark-based mobile
robot navigation: Guards are landmarks, which have to be distinguishable (hence
the colors), and are used to encode motion primitives, \eg, "move towards the
red landmark". Let , the chromatic number of , denote the minimum
number of colors required to color any guard cover of . We show that
determining, whether is \NP-hard for all . Keeping
the number of colors minimal is of great interest for robot navigation, because
less types of landmarks lead to cheaper and more reliable recognition
On the Computational Complexity of Defining Sets
Suppose we have a family of sets. For every , a
set is a {\sf defining set} for if is the
only element of that contains as a subset. This concept has been
studied in numerous cases, such as vertex colorings, perfect matchings,
dominating sets, block designs, geodetics, orientations, and Latin squares.
In this paper, first, we propose the concept of a defining set of a logical
formula, and we prove that the computational complexity of such a problem is
-complete.
We also show that the computational complexity of the following problem about
the defining set of vertex colorings of graphs is -complete:
{\sc Instance:} A graph with a vertex coloring and an integer .
{\sc Question:} If be the set of all -colorings of
, then does have a defining set of size at most ?
Moreover, we study the computational complexity of some other variants of
this problem
Volterra-assisted Optical Phase Conjugation: a Hybrid Optical-Digital Scheme For Fiber Nonlinearity Compensation
Mitigation of optical fiber nonlinearity is an active research field in the
area of optical communications, due to the resulting marked improvement in
transmission performance. Following the resurgence of optical coherent
detection, digital nonlinearity compensation (NLC) schemes such as digital
backpropagation (DBP) and Volterra equalization have received much attention.
Alternatively, optical NLC, and specifically optical phase conjugation (OPC),
has been proposed to relax the digital signal processing complexity. In this
work, a novel hybrid optical-digital NLC scheme combining OPC and a Volterra
equalizer is proposed, termed Volterra-Assisted OPC (VAO). It has a twofold
advantage: it overcomes the OPC limitation in asymmetric links and
substantially enhances the performance of Volterra equalizers. The proposed
scheme is shown to outperform both OPC and Volterra equalization alone by up to
4.2 dB in a 1000 km EDFA-amplified fiber link. Moreover, VAO is also
demonstrated to be very robust when applied to long-transmission distances,
with a 2.5 dB gain over OPC-only systems at 3000 km. VAO combines the
advantages of both optical and digital NLC offering a promising trade-off
between performance and complexity for future high-speed optical communication
systems
NP-hardness of circuit minimization for multi-output functions
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.
In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.
Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions
b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
A b-coloring of a graph is a proper coloring such that every color class
contains a vertex that is adjacent to all other color classes. The b-chromatic
number of a graph G, denoted by \chi_b(G), is the maximum number t such that G
admits a b-coloring with t colors. A graph G is called b-continuous if it
admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and
b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of
G, and every induced subgraph H_2 of H_1.
We investigate the b-chromatic number of graphs with stability number two.
These are exactly the complements of triangle-free graphs, thus including all
complements of bipartite graphs. The main results of this work are the
following:
- We characterize the b-colorings of a graph with stability number two in
terms of matchings with no augmenting paths of length one or three. We derive
that graphs with stability number two are b-continuous and b-monotonic.
- We prove that it is NP-complete to decide whether the b-chromatic number of
co-bipartite graph is at most a given threshold.
- We describe a polynomial time dynamic programming algorithm to compute the
b-chromatic number of co-trees.
- Extending several previous results, we show that there is a polynomial time
dynamic programming algorithm for computing the b-chromatic number of
tree-cographs. Moreover, we show that tree-cographs are b-continuous and
b-monotonic
- …