Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

Reinforcement-Learning-Based Quantum Adiabatic Algorithm Design for Integer Programming

Adiabatic quantum computing (AQC) is a computation protocol to solve difficult problems exploiting quantum advantage, directly applicable to optimization problems. In performing the AQC, different configurations of the Hamiltonian path could lead to dramatic differences in the computation efficiency. It is thus crucial to configure the Hamiltonian path to optimize the computation performance of AQC. Here we apply a reinforcement learning approach to configure AQC for integer programming, where we find the learning process automatically converges to a quantum algorithm that exhibits scaling advantage over the trivial AQC using a linear Hamiltonian path. This reinforcement-learning-based approach for quantum adiabatic algorithm design for integer programming can well be adapted to the quantum resources in different quantum computation devices, due to its built-in flexibility.

Eigenvalues, traceability, and perfect matching of a graph

It is well known that the recognition problem of the existence of a perfect matching in a graph, as well as the recognition problem of its Hamiltonicity and traceability, is NP-complete. Quite recently, lower bounds for the size and the spectral radius of a graph that guarantee the existence of a perfect matching in it have been obtained. We improve these bounds, firstly, by using the available bounds for the size of the graph for existence of a Hamiltonian path in it, and secondly, by finding new lower bounds for the spectral radius of the graph that are sufficient for the traceability property. Moreover, we develop the recognition algorithm of the existence of a perfect matching in a graph. This algorithm uses the concept of a (κ,τ)-regular set, which becomes polynomial in the class of graphs with a fixed cyclomatic number.

Symmetric Connectivity of Underwater Acoustic Sensor Networks Based on Multi-Modal Directional Transducer

Topology control is one of the most essential technologies in wireless sensor networks (WSNs); it constructs networks with certain characteristics through the usage of some approaches, such as power control and channel assignment, thereby reducing the inter-nodes interference and the energy consumption of the network. It is closely related to the efficiency of upper layer protocols, especially MAC and routing protocols, which are the same as underwater acoustic sensor networks (UASNs). Directional antenna technology (directional transducer in UASNs) has great advantages in minimizing interference and conserving energy by restraining the beamforming range. It enables nodes to communicate with only intended neighbors; nevertheless, additional problems emerge, such as how to guarantee the connectivity of the network. This paper focuses on the connectivity problem of UASNs equipped with tri-modal directional transducers, where the orientation of a transducer is stabilized after the network is set up. To efficiently minimize the total network energy consumption under constraint of connectivity, the problem is formulated to a minimum network cost transducer orientation (MNCTO) problem and is provided a reduction from the Hamiltonian path problem in hexagonal grid graphs (HPHGG), which is proved to be NP-complete. Furthermore, a heuristic greedy algorithm is proposed for MNCTO. The simulation evaluation results in a contrast with its omni-mode peer, showing that the proposed algorithm greatly reduces the network energy consumption by up to nearly half on the premise of satisfying connectivity.

Graph of Fuzzy Topographic Topological Mapping in relation to k-Fibonacci Sequence

A generated n-sequence of fuzzy topographic topological mapping, FTTM n , is a combination of n number of FTTM’s graphs. An assembly graph is a graph whereby its vertices have valency of one or four. A Hamiltonian path is a path that visits every vertex of the graph exactly once. In this paper, we prove that assembly graphs exist in FTTM n and establish their relations to the Hamiltonian polygonal paths. Finally, the relation between the Hamiltonian polygonal paths induced from FTTM n to the k-Fibonacci sequence is established and their upper and lower bounds’ number of paths is determined.

Computing directed Steiner path covers

In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph $$G=(V,E)$$ G = ( V , E ) and a set $$T \subseteq V$$ T ⊆ V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes

A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

JUMPER Enables Discontinuous Transcript Assembly in Coronaviruses

Genes in SARS-CoV-2 and other viruses in the order of Nidovirales are expressed by a process of discontinuous transcription mediated by the viral RNA-dependent RNA polymerase. This process is distinct from alternative splicing in eukaryotes and produces subgenomic RNAs that express different viral genes. Here, we introduce the DISCONTINUOUS TRANSCRIPT ASSEMBLY problem of finding transcripts T and their abundances c given an alignment R of paired end short reads under a maximum likelihood model that accounts for varying transcript lengths. Underpinning our approach is the concept of a segment graph, a directed acyclic graph that, distinct from the splice graph used to characterize alternative splicing, has a unique Hamiltonian path. We provide a compact characterization of solutions as subsets of non-overlapping edges in this graph, enabling the formulation of an efficient progressive heuristic that uses mixed integer linear program. We show using simulations that our method, JUMPER, drastically outperforms existing methods for classical transcript assembly. On short-read data of SARS-CoV-1, SARS-CoV-2 and MERS-CoV samples, we find that JUMPER not only identifies canonical transcripts that are part of the reference transcriptome, but also predicts expression of non-canonical transcripts that are well supported by direct evidence from long-read data, presence in multiple, independent samples or a conserved core sequence. Moreover, application of JUMPER on samples with and without treatment reveals viral drug response at the transcript level. As such, JUMPER enables detailed analyses of Nidovirales transcriptomes under varying conditions.