16,029 research outputs found
Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five
In this paper, we address the maximum number of vertices of induced forests
in subcubic graphs with girth at least four or five. We provide a unified
approach to prove that every 2-connected subcubic graph on vertices and
edges with girth at least four or five, respectively, has an induced forest on
at least or vertices, respectively, except
for finitely many exceptional graphs. Our results improve a result of Liu and
Zhao and are tight in the sense that the bounds are attained by infinitely many
2-connected graphs. Equivalently, we prove that such graphs admit feedback
vertex sets with size at most or , respectively.
Those exceptional graphs will be explicitly constructed, and our result can be
easily modified to drop the 2-connectivity requirement
Network of Time-Multiplexed Optical Parametric Oscillators as a Coherent Ising Machine
Finding the ground states of the Ising Hamiltonian [1] maps to various
combinatorial optimization problems in biology, medicine, wireless
communications, artificial intelligence, and social network. So far no
efficient classical and quantum algorithm is known for these problems, and
intensive research is focused on creating physical systems - Ising machines -
capable of finding the absolute or approximate ground states of the Ising
Hamiltonian [2-6]. Here we report a novel Ising machine using a network of
degenerate optical parametric oscillators (OPOs). Spins are represented with
above-threshold binary phases of the OPOs and the Ising couplings are realized
by mutual injections [7]. The network is implemented in a single OPO ring
cavity with multiple trains of femtosecond pulses and configurable mutual
couplings, and operates at room temperature. We programed the smallest
non-deterministic polynomial time (NP)- hard Ising problem on the machine, and
in 1000 runs of the machine no computational error was detected
Synchronously-pumped OPO coherent Ising machine: benchmarking and prospects
The coherent Ising machine (CIM) is a network of optical parametric oscillators (OPOs) that solves for the ground state of Ising problems through OPO bifurcation dynamics. Here, we present experimental results comparing the performance of the CIM to quantum annealers (QAs) on two classes of NP-hard optimization problems: ground state calculation of the Sherrington-Kirkpatrick (SK) model and MAX-CUT. While the two machines perform comparably on sparsely-connected problems such as cubic MAX-CUT, on problems with dense connectivity, the QA shows an exponential performance penalty relative to CIMs. We attribute this to the embedding overhead required to map dense problems onto the sparse hardware architecture of the QA, a problem that can be overcome in photonic architectures such as the CIM
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