Hamiltonian Decomposition Research Articles

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

On the total chromatic number of the direct product of cycles and complete graphs

A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either Δ(G) + 1 (called Type 1) or Δ(G) + 2 (called Type 2), where Δ(G) is the maximum degree of G. We consider the direct product of complete graphs Km × Kn. It is known that if at least one of the numbers m or n is even, then Km × Kn is Type 1, except for K2 × K2. We prove that the graph Km × Kn is Type 1 when both m and n are odd numbers, by using that the conformable condition is sufficient for the graph Km × Kn to be Type 1 when both m and n are large enough, and by constructing the target total colorings by using Hamiltonian decompositions and a specific color class, called guiding color. We additionally apply our technique to the direct product Cm × Kn of a cycle with a complete graph. Interestingly, we are able to find a Type 2 infinite family Cm × Kn, when m is not a multiple of 3 and n = 2. We provide evidence to conjecture that all other Cm × Kn are Type 1.

Assessment of various Hamiltonian partitionings for the electronic structure problem on a quantum computer using the Trotter approximation

Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts.

Depth analysis of variational quantum algorithms for the heat equation

Variational quantum algorithms are a promising tool for solving partial differential equations. The standard approach for its numerical solution is finite-difference schemes, which can be reduced to the linear algebra problem. We consider three approaches to solve the heat equation on a quantum computer. Using the direct variational method we minimize the expectation value of a Hamiltonian with its ground state being the solution of the problem under study. Typically, an exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speedup to be achieved. The Hadamard-test-based approach solves this problem, however, the performed simulations do not evidently prove that the Ansatz circuit has a polynomial depth with respect to the number of qubits. The Ansatz tree approach exploits an explicit form of the matrix that makes it possible to achieve an advantage over classical algorithms. In our numerical simulations with up to $n=11$ qubits, this method reveals the exponential speedup.

Nondipole signatures in ionization and high-order harmonic generation

We analyze nondipole effects arising in ionization and high-order harmonic generation for a two-dimensional hydrogen atom irradiated with either low- or high-frequency laser pulses. In the low-frequency case, the electron wave packet dynamics is dominated by rescattering processes within the laser pulse. Here both odd- and even-order harmonics are generated in the direction of the laser field propagation and polarization, respectively. For high-frequency pulses, such rescattering processes can be neglected. We demonstrate that a significant portion of photoelectrons is detected opposite to the laser pulse propagation direction as a consequence of their postpulse wave packet spreading and interaction with the parent ion. This is accompanied by rich interference structures formed in the momentum distributions of photoelectrons. Our results follow from the numerical solution of the time-dependent Schr\"odinger equation, which is based on the Suzuki-Trotter scheme with the split-step Fourier approach. The method relies on a Hamiltonian decomposition, where except for the components depending exclusively on the momentum or on the position operators, there are also terms depending on both momentum and position operators in particular configurations. We demonstrate that, as long as the latter does not depend on noncommuting coordinates of the momentum and position operators, nondipole effects in laser-matter interactions can be studied without applying extra approximations and unitary operations.

Rainbow subgraphs in Hamiltonian cycle decompositions of complete graphs

A Hamiltonian cycle decomposition of a complete graph K2n+1 is an n-edge-coloring of K2n+1 such that each color class is a Hamiltonian cycle. For a given Hamiltonian cycle decomposition, a subgraph of K2n+1 is called rainbow if all its edges are colored differently. Let H be any subgraph of K2n+1 consisting of n edges. In 1999, Wu conjectured that K2n+1 has a Hamiltonian cycle decomposition such that H is rainbow. In this paper, we show that the conjecture is true in the cases that |H|≤n+1, or H is a linear forest, or n≤5. Using this result, we partially confirm a conjecture on restricted size Ramsey numbers due to Miralaei and Shahsiah in 2020.

Backtracking Algorithms for Constructing the Hamiltonian Decomposition of a 4-Regular Multigraph

Backtracking Algorithms for Constructing the Hamiltonian Decomposition of a 4-Regular Multigraph

Better Adaptive Malicious Users Detection Algorithm in Human Contact Networks

A human contact network (HCN) consists of individuals moving around and interacting with each other. In HCN, it is essential to detect malicious users who break the data delivery through terminating the data delivery or tampering with the data. Since malicious users will pay more but gain less when breaking the data delivery of opportunistic contacts, we focus on the non-opportunistic contacts that occur more frequently and stably. It is observed that people contact with each other more frequently if they have more social features in common. In this paper, we build up topology structure for HCN based on social features, and propose a graph theoretical comparison detection model to perform malicious users detection. Then we present an adaptive detection scheme based on Hamiltonian cycle decomposition. Also, we define comparison-0-string and comparison-1-string to improve the detection efficiency. Moreover, we perform scenario simulations on real data to realize the detected process of malicious users. Experiments show that, when the number of malicious users is bounded by the dimension of HCN, our scheme has a detection rate of 100% with both false positive rate and false negative rate being 0%, and the running cost is also very low when compared to baseline approaches. When the number of malicious users exceeds the bound, the detection rate of our scheme decreases slowly, while the false positive rate and false negative rate increase slowly, but they are still better than the baseline approaches.

Hamiltonian decompositions of 4-regular Cayley graphs of infinite abelian groups.

A well‐known conjecture of Alspach says that every 2k‐regular Cayley graph of a finite abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double‐ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of 2k‐regular Cayley graphs with finite cuts F, where ∣F∣ and k differ in parity, which necessarily preclude the existence of a decomposition into spanning double‐rays. We show that every 4‐regular Cayley graph of an infinite abelian group all of whose finite cuts are even can be decomposed into spanning double‐rays, and so characterise when such decompositions exist. We also characterise when such graphs can be decomposed either into Hamiltonian circles, a more topological generalisation of a Hamiltonian cycle in infinite graphs, or into a Hamiltonian circle and a spanning double‐ray.

On Hamiltonian Decomposition Problem of 3-Arc Graphs.

A 4-tuple (y, x, v, w) in a graph is a 3-arc if each of (y, x, v) and (x, v, w) is a path. The 3-arc graph of H is the graph with vertex set all arcs of H and edge set containing all edges joining xy and vw whenever (y, x, v, w) is a 3-arc of H. A Hamilton cycle is a closed path meeting each vertex of a graph. A graph H including a Hamilton cycle is called Hamiltonian and H has a Hamiltonian decomposition provided its edge set admits a partition into disjoint Hamilton cycles (possibly with a single perfect matching). The current paper proves that every connected 3-arc graph consists of more than one Hamilton cycle. Since the 3-arc graph of a cubic graph is 4-regular, it further proves that each 3-arc graph of a cubic graph in a certain family has a Hamiltonian decomposition.

Embedding connected factorizations

For r:=(r1,…,rq), an r-factorization of the complete λ-fold h-uniform m-vertex hypergraph λKmh is a partition of the edges of λKmh into F1,…,Fq such that each color class Fi is ri-regular and spanning. We prove two results on embedding factorizations. Previously, these results were only known for a few small values of h, and even then only partially. We show that for n⩾hm and s:=(s1,…,sk), the obvious necessary conditions that ensure that an r-factorization of λKmh can be embedded into an s-factorization of λKnh are also sufficient. This extends Cruse's theorem, Baranyai's theorem, and Häggkvist-Hellgren's theorem. A connectedr-factorization is an r-factorization in which each color class is connected. For n⩾hm, we establish the necessary and sufficient conditions under which an r-factorization of λKmh can be embedded into a connected s-factorization of λKnh. This extends Walecki's theorem, and Hilton's theorem on embedding Hamiltonian decompositions (take λ=r1=…=rq=1,h=s1=…=sk=2).

A Hamiltonian decomposition for fast interior-point solvers in model predictive control

Optimal decision-making tools are essential in industry to achieve high performance. One of these tools is model predictive control (MPC), which is an advanced control technique that generates an action that affects the controlled variables, while satisfying the process’ operational constraints. At the core of the MPC algorithm lies an optimization problem that is solved by a numerical method at every sample time. New demand for more self-contained modular processes has seen MPC embedded in small-scale platforms. This has prompted a need for custom-made numerical methods that help to run the computationally demanding optimization algorithms efficiently. In this paper, we propose two approaches that factorize the Newton system of the interior-point method based on the two-point boundary-value problem structure, rarely explored in MPC. By exploiting the Hamiltonian form of the augmented system, we derive an incomplete LU factorization. A direct method is available to compute the solution of the system using a forward substitution of a series of matrices. We also propose a preconditioned Krylov method that converges within a small number of iterations only depending on the number of states.

Backtracking Algorithms for Constructing the Hamiltonian Decomposition of a 4-regular Multigraph

We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is an NP-complete problem. On the other hand, a suffcient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for verifying vertex non-adjacency in the 1-skeleton of the traveling salesperson polytope and constructing a Hamiltonian decomposition: an algorithm based on a simple path extension and an algorithm based on the chain edge fixing procedure. Based on the results of the computational experiments for undirected multigraphs, both backtracking algorithms lost to the known heuristic general variable neighborhood search algorithm. However, for directed multigraphs, the algorithm based on chain fixing of edges showed comparable results with heuristics on instances with existing solutions, and better results on instances of the problem where the Hamiltonian decomposition does not exist.

Hamiltonian decomposition and verifying vertex adjacency in 1-skeleton of the traveling salesperson polytope by variable neighborhood search

We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. We introduce a heuristic general variable neighborhood search algorithm for this problem based on finding a vertex-disjoint cycle cover of the multigraph through reduction to perfect matching and several cycle merging operations. The algorithm has a one-sided error: the answer "not adjacent" is always correct, and was tested on random directed and undirected Hamiltonian cycles and on pyramidal tours.

交换折叠交叉立方体的Hamilton分解及其性质

交换折叠交叉立方体的Hamilton分解及其性质

Compatible cycles and CHY integrals

The CHY construction naturally associates a vector in ℝ(n−3)! to every 2- regular graph with n vertices. Partial amplitudes in the biadjoint scalar theory are given by the inner product of vectors associated with a pair of cycles. In this work we study the problem of extending the computation to pairs of arbitrary 2-regular graphs. This requires the construction of compatible cycles, i.e. cycles such that their union with a 2-regular graph admits a Hamiltonian decomposition. We prove that there are at least (n − 2)!/4 such cycles for any 2-regular graph. We also find a connection to breakpoint graphs when the initial 2-regular graph only has double edges. We end with a comparison of the lower bound on the number of randomly selected cycles needed to generate a basis of ℝ(n−3)!, using the super Catalan numbers and our lower bound for compatible cycles.

Lifting of states in 2-dimensional N = 4 supersymmetric CFTs

We consider states of the D1-D5 CFT where only the left-moving sector is excited. As we deform away from the orbifold point, some of these states will remain BPS while others can ‘lift’. The lifting can be computed by a path integral containing two twist deformations; however, the relevant 4-point amplitude cannot be computed explicitly in many cases. We analyze an older proposal by Gava and Narain where the lift can be computed in terms of a finite number of 3-point functions. A direct Hamiltonian decomposition of the path integral involves an infinite number of 3-point functions, as well the first order correction to the starting state. We note that these corrections to the state account for the infinite number of 3-point functions arising from higher energy states, and one can indeed express the path-integral result in terms of a finite number of 3-point functions involving only the leading order states that are degenerate. The first order correction to the super-charge {overline{G}}^{(1)} gets replaced by a projection {overline{G}}^{(P)} ; this projected operator can also be used to group the states into multiplets whose members have the same lifting.

Embedding an edge‐coloring of K(nr;λ1,λ2) into a Hamiltonian decomposition of K(nr+2;λ1,λ2)

AbstractThis paper focuses on graph decompositions of , the ‐partite multigraph in which each part has size , where two vertices in the same part or different parts are joined by exactly edges or edges respectively. Assuming one condition, necessary and sufficient conditions are found to embed a k‐edge‐coloring of into a Hamiltonian decomposition of . In the tightest case, this assumption is in fact proved to be a new necessary condition. Unlike previous results, of particular interest here is a necessary condition involving the existence of certain components in a related bipartite graph.

Extended and Improved Criss-Cross Algorithms for Computing the Spectral Value Set Abscissa and Radius

In this paper, we extend the original criss-cross algorithms for computing the $\varepsilon$-pseudospectral abscissa and radius to general spectral value sets. By proposing new root-finding-based strategies for the horizontal/radial search subphases, we significantly reduce the number of expensive Hamiltonian eigenvalue decompositions incurred, which typically translates to meaningful speedups in overall computation times. Furthermore, and partly necessitated by our root-finding approach, we develop a new way of handling the singular pencils or problematic interior searches that can arise when computing the $\varepsilon$-spectral value set radius. Compared to would-be direct extensions of the original algorithms, that is, without our additional modifications, our improved criss-cross algorithms are not only noticeably faster but also more robust and numerically accurate, for both spectral value set and pseudospectral problems.

Proposal of the determination method of Lagrange multiplier using Hamiltonian decomposition of annealing optimization in Traveling Salesman Problem

Proposal of the determination method of Lagrange multiplier using Hamiltonian decomposition of annealing optimization in Traveling Salesman Problem