Efficient Algorithm to Determine whether a given Graph is Hamiltonian or Not with All Possible Paths

Given a Graph G (V, E), We Consider the problem of deciding whether G is Hamiltonian, that is- whether or Not there is a simple cycle in E spanning all vertices in V. [1] However to Verify that the given cycle is Hamiltonian by checking whether it is permutation of the vertices of V and whether each of the consecutives edges along the cycle actually exists in the Graph. This Verification Algorithm can certainly be implemented to run in O (n2) time, where n is the length of the encoding of G [2]. But to predict in Advance that the Graph has Hamiltonian Cycle or not was still Exponential before this Algorithm. This Problem is known to be NPComplete hence cannot be solved in Polynomial time in |V| unless P=NP. However till today there was no known Criterion we can apply to determine the existence Hamiltonian Circuit in General [3]. For its Exponential time We can Refer to theorems: - Vertex Cover problem is polynomially transformable to the Hamiltonian circuit Problem for Directed graphs, hence the Hamiltonian Circuit problem for Directed Graph is NP-Complete and the Hamiltonian Circuit Problem for Directed Graph is Polynomialy transformable to Hamiltonian Cycle Problem for Undirected Graph, hence the Hamiltonian Cycle Problem for undirected Graph is NP-complete [4]. Note that these derivations are based on the CNF- Satisfiability. Through this Paper we have introduced a Newer Algorithm with different approach to determine whether a given Graph is Hamiltonian or Not with all possible Paths, by applying Few Mathematical and logical Operations. This provides necessary and sufficient condition for a graph to be Hamiltonian

Quantum Optimization of Fully-Connected Spin Glasses

The Sherrington-Kirkpatrick model with random $\pm1$ couplings is programmed on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type graph. The performance of the optimizer compares and correlates to simulated annealing. When considering the effect of the static noise, which degrades the performance of the annealer, one can estimate an improvement on the comparative scaling of the two methods in favor of the D-Wave machine. The optimal choice of parameters of the embedding on the Chimera graph is shown to be associated to the emergence of the spin-glass critical temperature of the embedded problem.Comment: includes supplemental materia

Performance Models for Split-execution Computing Systems

Split-execution computing leverages the capabilities of multiple computational models to solve problems, but splitting program execution across different computational models incurs costs associated with the translation between domains. We analyze the performance of a split-execution computing system developed from conventional and quantum processing units (QPUs) by using behavioral models that track resource usage. We focus on asymmetric processing models built using conventional CPUs and a family of special-purpose QPUs that employ quantum computing principles. Our performance models account for the translation of a classical optimization problem into the physical representation required by the quantum processor while also accounting for hardware limitations and conventional processor speed and memory. We conclude that the bottleneck in this split-execution computing system lies at the quantum-classical interface and that the primary time cost is independent of quantum processor behavior.Comment: Presented at 18th Workshop on Advances in Parallel and Distributed Computational Models [APDCM2016] on 23 May 2016; 10 page