20 research outputs found
A set of tournaments with many Hamiltonian cycles
For a random tournament on vertices, the expected number of Hamiltonian cycles is known to be . Let denote a tournament of three vertices . Let the orientation be such that there are directed edges from to , from to and from to . Construct a tournament by making three copies of , , and . Let each vertex in have directed edges to all vertices in , similarly place directed edges from each vertex in to all vertices in and from to . In this thesis, we shall study this family of highly symmetric tournaments. In particular we shall present two different algorithms to calculate the number of Hamiltonian cycles in these tournaments and compare them with the expected number and with known bounds for random tournaments. This thesis is motivated by the question of the maximum number of Hamiltonian cycles a tournament can have
What Moser Could Have Asked: Counting Hamilton Cycles in Tournaments
Moser asked for a construction of explicit tournaments on vertices having
at least Hamilton cycles. We show that he could have asked
for rather more
Detecting Multiple Communities Using Quantum Annealing on the D-Wave System
A very important problem in combinatorial optimization is partitioning a
network into communities of densely connected nodes; where the connectivity
between nodes inside a particular community is large compared to the
connectivity between nodes belonging to different ones. This problem is known
as community detection, and has become very important in various fields of
science including chemistry, biology and social sciences. The problem of
community detection is a twofold problem that consists of determining the
number of communities and, at the same time, finding those communities. This
drastically increases the solution space for heuristics to work on, compared to
traditional graph partitioning problems. In many of the scientific domains in
which graphs are used, there is the need to have the ability to partition a
graph into communities with the ``highest quality'' possible since the presence
of even small isolated communities can become crucial to explain a particular
phenomenon. We have explored community detection using the power of quantum
annealers, and in particular the D-Wave 2X and 2000Q machines. It turns out
that the problem of detecting at most two communities naturally fits into the
architecture of a quantum annealer with almost no need of reformulation. This
paper addresses a systematic study of detecting two or more communities in a
network using a quantum annealer
Network Community Detection On Small Quantum Computers
In recent years a number of quantum computing devices with small numbers of
qubits became available. We present a hybrid quantum local search (QLS)
approach that combines a classical machine and a small quantum device to solve
problems of practical size. The proposed approach is applied to the network
community detection problem. QLS is hardware-agnostic and easily extendable to
new quantum computing devices as they become available. We demonstrate it to
solve the 2-community detection problem on graphs of size up to 410 vertices
using the 16-qubit IBM quantum computer and D-Wave 2000Q, and compare their
performance with the optimal solutions. Our results demonstrate that QLS
perform similarly in terms of quality of the solution and the number of
iterations to convergence on both types of quantum computers and it is capable
of achieving results comparable to state-of-the-art solvers in terms of quality
of the solution including reaching the optimal solutions
Quantum Local Search for Graph Community Detection
We present Quantum Local Search (QLS) approach and demonstrate its efficacy by applying it to the problem of community detection in real-world networks. QLS is a hybrid algorithm that combines a classical machine with a small quantum device. QLS starts with an initial solution and searches its neighborhood, iteratively trying to find a better candidate solution. One of the main challenges of the quantum computing in NISQ era is the small number of available qubits. QLS addresses this challenge by using the quantum device only for the neighborhood search, which can be restricted to be small enough to fit on near-term quantum device. We implement QLS for modularity maximization graph clustering using QAOA on IBM Q Experience as a quantum local solver. We demonstrate the potential for quantum acceleration by showing that existing state-of-the-art optimization solvers cannot find a good solution to the local problems quickly and provide an estimate of how larger quantum devices can improve the performance of QLS