H-colorings are essential in the exploration of language syntax expositions. An H-coloring is the allocation of adjacent colors to neighboring vertices of a graph G. An H-coloring is a homomorphism mapping of a graph G to H. Even though the computational intricacy for calculating the precise solutions to count the lists of H-colorings exists in the literature, questions remains as to how the approximate count should be derived. Galanis and colleagues present a complexity trichotomy for practically enumerating the H-colorings of a graph.
A fully polynomial randomized approximation scheme is a randomized technique for generating foreseeable problem solutions within a definite comparative error, with high likelihood in polynomial time. An approximate preserving reduction is a randomized Turing reduction for computing close solution approximations of a problem, given the close approximations of any other problem. #SAT is the enumeration problem that satisfies the assignments of a Boolean formula. #BIS is the problem of enumerating autonomous sets in a bipartite graph.
Given a connected undirected loop-free or circling graph H, the complexity trichotomy creators theorized and verified that (1) for a reflexive all-inclusive or an irreflexive two-part graph H and a graph G with utmost degree vertices, the H-colorings can be enumerated in polynomial time, and (2) for a reflexive appropriate interval or an irreflexive two-part permutation graph H and a graph G with highest degree vertices of at least six, the H-colorings enumeration can be approximated from the autonomous sets in a bipartite graph.
The authors convincingly state and prove lemmas for #SAT-equivalent, #BIS-hard, and #BIS-easy problems that were used to ascertain the complexity trichotomy theorems. They skillfully use matrix algebra and examples of irreflexive non-bipartite permutation graphs and reflexive improper interval graphs to illuminate the proofs of the lemmas for approximately counting the list of H-colorings.
I strongly recommend that all researchers in the areas of formal languages and theory of computing read the insightful approaches to the proofs of lemmas and theorems in this paper.