Related papers: A Memetic Algorithm To Find a Hamiltonian Cycle in…
We are motivated by a tantalizing open question in exact algorithms: can we detect whether an $n$-vertex directed graph $G$ has a Hamiltonian cycle in time significantly less than $2^n$? We present new randomized algorithms that improve…
We introduce the problem of hidden Hamiltonian cycle recovery, where there is an unknown Hamiltonian cycle in an $n$-vertex complete graph that needs to be inferred from noisy edge measurements. The measurements are independent and…
A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltonian cycles in a given regular graph has many applications in combinatorial optimization and…
We present a matching and LP based heuristic algorithm that decides graph non-Hamiltonicity. Each of the $n!$ Hamilton cycles in a complete directed graph on $n+1$ vertices corresponds with each of the $n!$ $n$-permutation matrices $P$,…
We present an algorithm CRE, which either finds a Hamilton cycle in a graph $G$ or determines that there is no such cycle in the graph. The algorithm's expected running time over input distribution $G\sim G(n,p)$ is $(1+o(1))n/p$, the…
We train a small message-passing graph neural network to predict Hamiltonian cycles on Erd\H{o}s-R\'enyi random graphs in a critical regime. It outperforms existing hand-crafted heuristics after about 2.5 hours of training on a single GPU.…
The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost Hamilton cycle in an edge-weighted graph, where costs are defined on all pairs of edges contained in the Hamilton cycle. This is a more general version than the…
The Hamiltonian cycle (HC) problem in graph theory is a well-known NP-complete problem. We present an approach in terms of $\mathbb{Z}_2$ lattice gauge theory (LGT) defined on the lattice with the graph as its dual. When the coupling…
The Hamiltonian cycle polynomial can be evaluated to count the number of Hamiltonian cycles in a graph. It can also be viewed as a list of all spanning cycles of length $n$. We adopt the latter perspective and present a pair of original…
We study a certain polytope arising from embedding the Hamiltonian cycle problem in a discounted Markov decision process. The Hamiltonian cycle problem can be reduced to finding particular extreme points of a certain polytope associated…
The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…
We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least $n/2$, where $n$ denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs…
The construction of the random intersection graph model is based on a random family of sets. Such structures, which are derived from intersections of sets, appear in a natural manner in many applications. In this article we study the…
We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the…
We revisit the method of small subgraph conditioning, used to establish that random regular graphs are Hamiltonian a.a.s. We refine this method using new technical machinery for random $d$-regular graphs on $n$ vertices that hold not just…
We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $\delta(H)\geq\alpha n$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph,…
Hamiltonian Monte Carlo (HMC) algorithms which combine numerical approximation of Hamiltonian dynamics on finite intervals with stochastic refreshment and Metropolis correction are popular sampling schemes, but it is known that they may…
A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It serves as a model of a compact polymer on a lattice. I study the number of Hamiltonian cycles, or equivalently the entropy of a compact polymer,…
We present fast and efficient randomized distributed algorithms to find Hamiltonian cycles in random graphs. In particular, we present a randomized distributed algorithm for the $G(n,p)$ random graph model, with number of nodes $n$ and…
We present a benchmark set for Traveling salesman problem (TSP) with characteristics that are different from the existing benchmark sets. In particular, we focus on small instances which prove to be challenging for one or more…