1,183 research outputs found
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Scalable partitioning for parallel position based dynamics
We introduce a practical partitioning technique designed for parallelizing Position Based Dynamics, and exploiting
the ubiquitous multi-core processors present in current commodity GPUs. The input is a set of particles whose
dynamics is influenced by spatial constraints. In the initialization phase, we build a graph in which each node
corresponds to a constraint and two constraints are connected by an edge if they influence at least one common
particle. We introduce a novel greedy algorithm for inserting additional constraints (phantoms) in the graph
such that the resulting topology is q-colourable, where ˆ qˆ ≥ 2 is an arbitrary number. We color the graph, and
the constraints with the same color are assigned to the same partition. Then, the set of constraints belonging to
each partition is solved in parallel during the animation phase. We demonstrate this by using our partitioning
technique; the performance hit caused by the GPU kernel calls is significantly decreased, leaving unaffected the
visual quality, robustness and speed of serial position based dynamics
RASCAL: calculation of graph similarity using maximum common edge subgraphs
A new graph similarity calculation procedure is introduced for comparing labeled graphs. Given a minimum similarity threshold, the procedure consists of an initial screening process to determine whether it is possible for the measure of similarity between the two graphs to exceed the minimum threshold, followed by a rigorous maximum common edge subgraph (MCES) detection algorithm to compute the exact degree and composition of similarity. The proposed MCES algorithm is based on a maximum clique formulation of the problem and is a significant improvement over other published algorithms. It presents new approaches to both lower and upper bounding as well as vertex selection
Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures
Quantum computers have recently made great strides and are on a long-term
path towards useful fault-tolerant computation. A dominant overhead in
fault-tolerant quantum computation is the production of high-fidelity encoded
qubits, called magic states, which enable reliable error-corrected computation.
We present the first detailed designs of hardware functional units that
implement space-time optimized magic-state factories for surface code
error-corrected machines. Interactions among distant qubits require surface
code braids (physical pathways on chip) which must be routed. Magic-state
factories are circuits comprised of a complex set of braids that is more
difficult to route than quantum circuits considered in previous work [1]. This
paper explores the impact of scheduling techniques, such as gate reordering and
qubit renaming, and we propose two novel mapping techniques: braid repulsion
and dipole moment braid rotation. We combine these techniques with graph
partitioning and community detection algorithms, and further introduce a
stitching algorithm for mapping subgraphs onto a physical machine. Our results
show a factor of 5.64 reduction in space-time volume compared to the best-known
previous designs for magic-state factories.Comment: 13 pages, 10 figure
Solving constraint-satisfaction problems with distributed neocortical-like neuronal networks
Finding actions that satisfy the constraints imposed by both external inputs
and internal representations is central to decision making. We demonstrate that
some important classes of constraint satisfaction problems (CSPs) can be solved
by networks composed of homogeneous cooperative-competitive modules that have
connectivity similar to motifs observed in the superficial layers of neocortex.
The winner-take-all modules are sparsely coupled by programming neurons that
embed the constraints onto the otherwise homogeneous modular computational
substrate. We show rules that embed any instance of the CSPs planar four-color
graph coloring, maximum independent set, and Sudoku on this substrate, and
provide mathematical proofs that guarantee these graph coloring problems will
convergence to a solution. The network is composed of non-saturating linear
threshold neurons. Their lack of right saturation allows the overall network to
explore the problem space driven through the unstable dynamics generated by
recurrent excitation. The direction of exploration is steered by the constraint
neurons. While many problems can be solved using only linear inhibitory
constraints, network performance on hard problems benefits significantly when
these negative constraints are implemented by non-linear multiplicative
inhibition. Overall, our results demonstrate the importance of instability
rather than stability in network computation, and also offer insight into the
computational role of dual inhibitory mechanisms in neural circuits.Comment: Accepted manuscript, in press, Neural Computation (2018
Contemporary Methods for Graph Coloring as an Example of Discrete Optimization
This paper provides an insight into graph coloringapplication of the contemporary heuristic methods. It discusses avariety of algorithmic solutions for The Graph Coloring Problem(GCP) and makes recommendations for implementation. TheGCP is the NP-hard problem, which aims at finding the minimumnumber of colors for vertices in such a way, that none of twoadjacent vertices are marked with the same color.With the adventof multicore processing technology, the metaheuristic approachto solving GCP reemerged as means of discrete optimization. Toexplain the phenomenon of these methods, the author makes athorough survey of AI-based algorithms for GCP, while pointingout the main differences between all these techniques
Finding Optimal 2-Packing Sets on Arbitrary Graphs at Scale
A 2-packing set for an undirected graph is a subset such that any two vertices have no common
neighbors. Finding a 2-packing set of maximum cardinality is a NP-hard problem.
We develop a new approach to solve this problem on arbitrary graphs using its
close relation to the independent set problem. Thereby, our algorithm red2pack
uses new data reduction rules specific to the 2-packing set problem as well as
a graph transformation. Our experiments show that we outperform the
state-of-the-art for arbitrary graphs with respect to solution quality and also
are able to compute solutions multiple orders of magnitude faster than
previously possible. For example, we are able to solve 63% of our graphs to
optimality in less than a second while the competitor for arbitrary graphs can
only solve 5% of the graphs in the data set to optimality even with a 10 hour
time limit. Moreover, our approach can solve a wide range of large instances
that have previously been unsolved
Contemporary Methods for Graph Coloring as an Example of Discrete Optimization
This paper provides an insight into graph coloringapplication of the contemporary heuristic methods. It discusses avariety of algorithmic solutions for The Graph Coloring Problem(GCP) and makes recommendations for implementation. TheGCP is the NP-hard problem, which aims at finding the minimumnumber of colors for vertices in such a way, that none of twoadjacent vertices are marked with the same color.With the adventof multicore processing technology, the metaheuristic approachto solving GCP reemerged as means of discrete optimization. Toexplain the phenomenon of these methods, the author makes athorough survey of AI-based algorithms for GCP, while pointingout the main differences between all these techniques
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