Deciding Graph non-Hamiltonicity via a Closure Algorithm

TL;DR

This paper introduces a heuristic algorithm based on matching and LP techniques to determine graph non-Hamiltonicity, demonstrating effectiveness on various non-Hamiltonian graphs and providing insights into Hamiltonian cycle structures.

Contribution

The paper proposes a novel closure algorithm combining matching and LP methods to decide graph non-Hamiltonicity, advancing computational approaches in this problem.

Findings

Successfully identified non-Hamiltonian graphs with up to 104 vertices.

Correctly decided non-Hamiltonian status for 2000 randomized 31-vertex graphs.

Presented cases where the algorithm remains undecided, indicating limitations.

Abstract

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 $P$, such that $p_{u,i}=1$ if and only if the $i^{th}$ arc in a cycle enters vertex $u$, starting and ending at vertex $n+1$. A graph instance ($G$) is initially coded as exclusion set $E$, whose members are pairs of components of $P$, $\{p_{u,i} ,p_{v,i+1}\}, i=1,n-1$, for each arc $(u,v)$ not in $G$. For each $\{p_{u,i} ,p_{v,i+1}\}\in E$, the set of $P$ satisfying $p_{u,i}=p_{v,i+1}=1$ correspond with a set of cycles not in $G$. Accounting for all arcs not in $G$, $E$ codes precisely the set of cycles not in $G$. A doubly stochastic-like $\mathcal{O}$($n^4$) formulation of the Hamilton cycle decision problem is then constructed. Each $\{p_{u,i} ,p_{v,j}\}$ is coded as variable $q_{u,i,v,j}$ such that the set of integer extrema is the set of all permutations. We model $G$ by setting each $q_{u,i,v,j}=0$ in correspondence with each $\{p_{u,i} ,p_{v,j}\}\in E$ such that for non-Hamiltonian $G$, integer solutions cannot exist. We recognize non-Hamiltonicity by iteratively deducing additional $q_{u,i,v,j}$ that can be set zero and expanding $E$ until the formulation becomes infeasible, in which case we recognize that no integer solutions exists i.e. $G$ is decided non-Hamiltonian. Over 100 non-Hamiltonian graphs (10 through 104 vertices) and 2000 randomized 31 vertex non-Hamiltonian graphs are tested and correctly decided non-Hamiltonian. For Hamiltonian $G$, the complement of $E$ provides information about covers of matchings, perhaps useful in searching for cycles. We also present an example where the algorithm fails to deduce any integral value for any $q_{u,i,v,j}$ i.e. $G$ is undecided.