You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
We consider the subgraph isomorphism problem where, given two graphs G (source graph) and F (pattern graph), one is to decide whether there is a (not necessarily induced) subgraph of G isomorphic to F. While many practical heuristic algorithms have been developed for the problem, as pointed out by [JAIR 2018], for each of them there are rather small instances which they cannot cope. Therefore, developing an alternative approach that could possibly cope with these hard instances would be of interest. A seminal paper by Alon, Yuster and Zwick [J. ACM 1995] introduced the color coding approach to solve the problem, where the main part is a dynamic programming over color subsets and partial mappings. As with many exponential-time dynamic programming algorithms, the memory requirements constitute the main limiting factor for its usage. Because these requirements grow exponentially with the treewidth of the pattern graph, all existing implementations based on the color coding principle restrict themselves to specific pattern graphs, e.g., paths or trees. In contrast, we provide an efficient implementation of the algorithm significantly reducing its memory requirements so that it can be used for pattern graphs of larger treewidth. Moreover, our implementation not only decides the existence of an isomorphic subgraph, but it also enumerates all such subgraphs (or given number of them). We provide an extensive experimental comparison of our implementation to other available solvers for the problem.

Cited Abstract(s):
[1]: Graphs are a powerful data structure that can be applied to several problems in bioinformatics. Graph matching, in its diverse forms, is an important operation on graphs, involved when there is the need to compare two graphs or to find substructures into larger structures. Many graph matching algorithms exist, and their relative efficiency depends on the kinds of graphs they are applied to. In this paper we will consider some popular and freely available matching algorithms, and will experimentally compare them on graphs derived from bioinformatics applications, in order to help the researchers in this field to choose the right tool for the problem at hand.

[2]: Subgraph isomorphism is a computationally challenging problem with important practical applications, for example in computer vision, biochemistry, and model checking. There are a number of state-of-the-art algorithms for solving the problem, each of which has its own performance characteristics. As with many other hard problems, the single best choice of algorithm overall is rarely the best algorithm on an instance-by-instance. We develop an algorithm selection approach which leverages novel features to characterise subgraph isomorphism problems and dynamically decides which algorithm to use on a per-instance basis. We demonstrate substantial performance improvements on a large set of hard benchmark problems. In addition, we show how algorithm selection models can be leveraged to gain new insights into what affects the performance of an algorithm.

[3]: The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. The non-induced version allows extra edges in the target, whilst the induced version does not. Although both variants are NP-complete, algorithms inspired by constraint programming can operate comfortably on many real-world problem instances with thousands of vertices. However, they cannot handle arbitrary instances of this size. We show how to generate " really hard " random instances for subgraph isomorphism problems, which are computationally challenging with a couple of hundred vertices in the target, and only twenty pattern vertices. For the non-induced version of the problem, these instances lie on a satisfiable unsatisfiable phase transition, whose location we can predict; for the induced variant, much richer behaviour is observed, and constrained-ness gives a better measure of difficulty than does proximity to a phase transition. These results have practical consequences: we explain why the widely researched " filter verify " indexing technique used in graph databases is founded upon a misunderstanding of the empirical hardness of NP-complete problems, and cannot be beneficial when paired with any reasonable subgraph isomorphism algorithm.
Our aim is to make step towards competitive implementation of color coding based algorithm for , in order to see, where this approach can be potentially beneficial against the existing algorithms. To this end, we extend the comparisons of algorithms [1] [2] [3] to color coding based algorithms, including the one proposed in this paper.