Hamiltonian Path Problem Research Articles

Solving QUBO problems with cP systems

P systems with compound terms (cP systems) have been proposed by Radu Nicolescu in 2018. These expressive cP systems have been used to solve well-known NP-complete problems efficiently, such as the Hamiltonian path, traveling salesman, 3-coloring, and software verification problems. In this paper, we use cP systems to provide an efficient parallel solution to the integer-valued quadratic unconstrained Boolean optimization (QUBO) problem.

GLOP: Learning Global Partition and Local Construction for Solving Large-Scale Routing Problems in Real-Time

The recent end-to-end neural solvers have shown promise for small-scale routing problems but suffered from limited real-time scaling-up performance. This paper proposes GLOP (Global and Local Optimization Policies), a unified hierarchical framework that efficiently scales toward large-scale routing problems. GLOP hierarchically partitions large routing problems into Travelling Salesman Problems (TSPs) and TSPs into Shortest Hamiltonian Path Problems. For the first time, we hybridize non-autoregressive neural heuristics for coarse-grained problem partitions and autoregressive neural heuristics for fine-grained route constructions, leveraging the scalability of the former and the meticulousness of the latter. Experimental results show that GLOP achieves competitive and state-of-the-art real-time performance on large-scale routing problems, including TSP, ATSP, CVRP, and PCTSP. Our code is available at: https://github.com/henry-yeh/GLOP.

An extension of the Christofides heuristic for a single-depot multiple Hamiltonian path problem

An extension of the Christofides heuristic for a single-depot multiple Hamiltonian path problem

Beyond Quasi-Adjoint Graphs: On Polynomial-Time Solvable Cases of the Hamiltonian Cycle and Path Problems

The Hamiltonian cycle and path problems are fundamental in graph theory and useful in modelling real-life problems. Research in this area is directed toward designing better and better algorithms for general problems, but also toward defining new special cases for which exact polynomial-time algorithms exist. In the paper, such new classes of digraphs are proposed. The classes include, among others, quasi-adjoint graphs, which are a superclass of adjoints, directed line graphs, and graphs modelling a DNA sequencing problem.

Minor embedding in broken chimera and derived graphs is NP-complete

The embedding is an essential step when calculating on the D-Wave machine. In this work, we show the hardness of the embedding problem for all types of currently existing hardware, represented by the Chimera and the derived Pegasus and Zephyr graphs, containing unavailable qubits. We construct certain broken Chimera graphs, where it is hard to find a Hamiltonian cycle. As the Hamiltonian cycle problem is a special case of the embedding problem, this proves the general complexity result for the Chimera graphs. By exploiting the subgraph relation between the Chimera and the derived graphs, the proof is then further extended to the Pegasus graphs and to the Zephyr graphs.

Multiagent, Multitarget Path Planning in Markov Decision Processes

Missions for autonomous systems often require agents to visit multiple targets in complex operating conditions. This work considers the problem of visiting a set of targets in minimum time by a team of non-communicating agents in a Markov decision process (MDP). The single-agent problem is at least NP-complete by reducing it to a Hamiltonian path problem. We first discuss an optimal algorithm based on Bellman's optimality equation that is exponential in the number of target states. Then, we trade-off optimality for time complexity by presenting a suboptimal algorithm that is polynomial at each time step. We prove that the proposed algorithm generates optimal policies for certain classes of MDPs. Extending our procedure to the multi-agent case, we propose a target partitioning algorithm that approximately minimizes the expected time to visit the targets. We prove that our algorithm generates optimal partitions for clustered target scenarios. We present the performance of our algorithms on random MDPs and gridworld environments inspired by ocean dynamics. We show that our algorithms are much faster than the optimal procedure and more optimal than the currently available heuristic.

Correspondence between the Hamiltonian cycle problem and the quantum lattice gauge theory

We propose the correspondence between the Hamiltonian cycle (HC) problem in graph theory and the quantum lattice gauge theory (QZ2LGT) defined on the lattice dual to that graph. For the QZ2LGT, when the coupling parameter g is less than the critical value g c , the ground state is a superposition of all configurations with closed strings of same spins, which can be obtained by using an adiabatic quantum algorithm. A subsequent search for a HC among those closed strings solves the original HC problem. The method is demonstrated for random samples of small graphs.

A method of automatic field of view generation and path planning for automated x‐ray inspection

AbstractWith the increasing accuracy and stability requirements of SMT equipment, automatic x‐ray inspection (AXI) is used as a new popular type of testing technology. First, an imaging constraint model is proposed to ensure that the detected object is not truncated. Second, an improved iterative self‐organizing clustering algorithm was proposed to realize an automatic optimal generation of the field of view and form a Hamiltonian path. Finally, the solution of the shortest Hamiltonian path problem is given by the proposed VTSP heuristic algorithm and VTSP + transformer deep learning algorithm. Experimental results show that, compared with the binary state compression DP method and the local shortest path planning method, the proposed method can support large‐scale nodes, and the performance is improved by 16% on 75 nodes and 31% on 200 nodes. The proposed method achieves a balance between x‐ray imaging efficiency and performance.

Phase Transitions in Bandits with Switching Constraints

We consider the classic stochastic multiarmed bandit problem with a constraint that limits the total cost incurred by switching between actions to be no larger than a given switching budget. For this problem, we prove matching upper and lower bounds on the optimal (i.e., minimax) regret and provide efficient rate-optimal algorithms. Surprisingly, the optimal regret of this problem exhibits a nonconventional growth rate in terms of the time horizon and the number of arms. Consequently, we discover surprising “phase transitions” regarding how the optimal regret rate changes with respect to the switching budget: when the number of arms is fixed, there are equal-length phases, in which the optimal regret rate remains (almost) the same within each phase and exhibits abrupt changes between phases; when the number of arms grows with the time horizon, such abrupt changes become subtler and may disappear, but a generalized notion of phase transitions involving certain new measurements still exists. The results enable us to fully characterize the trade-off between the regret rate and the incurred switching cost in the stochastic multiarmed bandit problem, contributing new insights to this fundamental problem. Under the general switching cost structure, the results reveal interesting connections between bandit problems and graph traversal problems, such as the shortest Hamiltonian path problem. This paper was accepted by J. George Shanthikumar, data science. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4755 .

A 3/2-approximation algorithm for the multiple Hamiltonian path problem with no prefixed endpoints

We study the multiple Hamiltonian path problem (MHPP) defined on a complete undirected graph G with n vertices. The edge weights of G are non-negative and satisfy the triangle inequality. The MHPP seeks to find a collection of k paths with exactly one visit to each vertex of G with the minimum total edge weight, where endpoints of the paths are not prefixed. We present a 3/2-approximation algorithm for MHPP with time complexity O(n3) for arbitrary k≥1.

AlphaSnake: Policy Iteration on a Nondeterministic NP-Hard Markov Decision Process (Student Abstract)

Reinforcement learning has been used to approach well-known NP-hard combinatorial problems in graph theory. Among these, Hamiltonian cycle problems are exceptionally difficult to analyze, even when restricted to individual instances of structurally complex graphs. In this paper, we use Monte Carlo Tree Search (MCTS), the search algorithm behind many state-of-the-art reinforcement learning algorithms such as AlphaZero, to create autonomous agents that learn to play the game of Snake, a game centered on properties of Hamiltonian cycles on grid graphs. The game of Snake can be formulated as a single-player discounted Markov Decision Process (MDP), where the agent must behave optimally in a stochastic environment. Determining the optimal policy for Snake, defined as the policy that maximizes the probability of winning -- or win rate -- with higher priority and minimizes the expected number of time steps to win with lower priority, is conjectured to be NP-hard. Performance-wise, compared to prior work in the Snake game, our algorithm is the first to achieve a win rate over 0.5 (a uniform random policy achieves a win rate < 2.57 x 10^{-15}), demonstrating the versatility of AlphaZero in tackling NP-hard problems.

Counting Traversing Hamiltonian Cycles in Tiled Graphs

Recently, the problem of counting Hamiltonian cycles in 2-tiled graphs was resolved by Vegi Kalamar, Bokal, and Žerak. In this paper, we continue our research on generalized tiled graphs. We extend algorithms on counting traversing Hamiltonian cycles from 2-tiled graphs to generalized tiled graphs. We further show that, similarly as for 2-tiled graphs, for a fixed finite set of tiles, counting traversing Hamiltonian cycles can be performed in linear time with respect to the size of such graph, implying that counting traversing Hamiltonian cycles in tiled graphs is fixed-parameter tractable.

CMSTR: A Constrained Minimum Spanning Tree Based Routing Protocol for Wireless Sensor Networks

How to extend the network lifetime with given limited energy budget is always one of the main concerns in Wireless Sensor Networks (WSNs). However, imbalanced energy consumption and overlong intra-cluster communication paths are prevalent in the hierarchical routing protocols, which shortens the network lifetime inevitably. To this end, an energy-efficient routing Protocol based on Constrained Minimum Spanning Tree (CMSTR) is proposed in this paper. To be specific, a new multichain routing scheme to balance the energy consumption for intra-cluster communications is presented. Based on the multichain routing scheme, the problem of establishing intra-cluster routing is transformed into a shortest Hamiltonian path problem on the basis of a graph-theoretic analysis model, which is solved through a Constrained Minimum Spanning Tree (CMST) algorithm proposed in this paper, with the aim to obtain the initial path for intra-cluster communications. In order to shorten the initial path length to obtain higher energy-efficient chain routes, a Neighbor Node Replacement (NNR) algorithm and a Link Intersection Detection and Elimination (LIDE) algorithm are proposed, in which the problem of potential long links and intersections is to be effectively alleviated. With shorter chain routes, unnecessary intra-cluster communication energy depletion can be reduced accordingly. In order to evaluate the performance of CMSTR, extensive simulation experiments are conducted. The results show that CMSTR can greatly prolong the network lifetime with regard to the metrics of FND and HND. To be specific, compared with LEACH, R-LEACH, and DCMSTR, the value of FND increased by 800%, 540% and 57%, that of HND increased by 322%, 286% and 22%, and overall network lifetime (AND) increased by 29%, 10% and 5%, respectively. Besides, CMSTR has a stable and lowest packet loss percentage (0.4%). In summary, CMSTR has excellent performance in terms of energy efficiency and network stability.

Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs

Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case.

A linear-time algorithm for finding Hamiltonian cycles in rectangular grid graphs with two rectangular holes

The Hamiltonian cycle problem is one of the most important problems in graph theory that has many applications. This problem is NP-complete for general grid graphs. For solid grid graphs, there are polynomial-time algorithms. Existence of polynomial-time algorithms for grid graphs with few holes has been asked in the literature. In this paper, we give a linear-time algorithm for rectangular grid graphs with two rectangular holes. This extends the previous result for rectangular grid graphs with one rectangular hole. We first present the forbidden conditions in which there is no Hamiltonian cycle for any grid graphs, including rectangular grid graphs with rectangular holes. We then show that if these forbidden conditions do not hold, then there exists a Hamiltonian cycle. The proof is constructive, therefore, it gives an algorithm. An application of the problem is in off-line robot exploration.

SFC Design and VNF Placement Based on Traffic Volume Scaling and VNF Dependency in 5G Networks

The development of Fifth-Generation (5G) mobile communication technology has remarkably promoted the spread of the Internet of Things (IoT) applications. As a promising paradigm for IoT, edge computing can process the amount of data generated by mobile intelligent devices in less time response. Network Function Virtualization (NFV) that decouples network functions from dedicated hardware is an important architecture to implement edge computing, deploying heterogeneous Virtual Network Functions (VNF) (such as computer vision, natural language processing, intelligent control, etc.) on the edge service nodes. With the NFV MANO (Management and Orchestration) framework, a Service Function Chain (SFC) that contains a set of ordered VNFs can be constructed and placed in the network to offer a customized network service. However, the procedure of NFV orchestration faces a technical challenge in minimizing the network cost of VNF placement due to the complexity of the changing effect of traffic volume and the dependency on the VNF relationship. To this end, we jointly optimize SFC design and VNF placement to minimize resource cost while taking account of VNF dependency and traffic volume scaling. First, the problem is formulated as an Integer Linear Programming (ILP) model and proved NP-hard by reduction from Hamiltonian Cycle problem. Then we proposed an efficient heuristic algorithm called Traffic Aware and Interdependent VNF Placement (TAIVP) to solve the problem. Compared with the benchmark algorithms, emulation results show that our algorithm can reduce network cost by 10.2% and increase service request acceptance rate by 7.6% on average.

Comparison among Classical, Probabilistic and Quantum Algorithms for Hamiltonian Cycle Problem

Comparison among Classical, Probabilistic and Quantum Algorithms for Hamiltonian Cycle Problem

Asymmetric probabilistic minimum-cost Hamiltonian cycle problem considering arc and vertex failures

In this paper, we introduce the asymmetric probabilistic minimum-cost Hamiltonian cycle problem (APMCHCP) where arcs and vertices in the graph are possible to fail. APMCHCP is a generalization of traditional asymmetric minimum-cost Hamiltonian cycle problem, and it has applications in many emerging areas, such as post-disaster recovery, electronic circuit design, and security maintenance of wireless sensor networks. For each vertex, we define a chance-constraint to guarantee that the probability of arriving at the vertex must be greater than or equal to a given threshold. Four mixed-integer programming (MIP) formulations are proposed for modeling the problem, including two direct formulations and two recursive formulations. A combinatorial branch-and-bound (CBB) algorithm is proposed for solving the APMCHCP, where data preprocessing steps, feasibility rules, and approaches of finding upper and lower bounds are developed. In the numerical experiments, the CBB algorithm is compared with a commercial MIP solver (Gurobi 8.0) on a test-bed of two popular benchmark instance sets. The results show that the proposed CBB algorithm significantly outperforms Gurobi solver in terms of both the size of optimally solved instances and the computing time.

Finding Hamiltonian cycles of truncated rectangular grid graphs in linear time

The Hamiltonian cycle problem is an important problem in graph theory. For solid grid graphs, an O(n4)-time algorithm has been given. In this paper, we solve the problem for a special class of solid grid graphs, i.e. truncated rectangular grid graphs, in linear time.

Algorithms for maximum internal spanning tree problem for some graph classes

For a given graph G, a maximum internal spanning tree of G is a spanning tree of G with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.