Sketching methods with small window guarantee using minimum decycling sets

Most sequence sketching methods work by selecting specific $k$-mers from sequences so that the similarity between two sequences can be estimated using only the sketches. Because estimating sequence similarity is much faster using sketches than using sequence alignment, sketching methods are used to reduce the computational requirements of computational biology software packages. Applications using sketches often rely on properties of the $k$-mer selection procedure to ensure that using a sketch does not degrade the quality of the results compared with using sequence alignment. Two important examples of such properties are locality and window guarantees, the latter of which ensures that no long region of the sequence goes unrepresented in the sketch. A sketching method with a window guarantee, implicitly or explicitly, corresponds to a Decycling Set, an unavoidable sets of $k$-mers. Any long enough sequence, by definition, must contain a $k$-mer from any decycling set (hence, it is unavoidable). Conversely, a decycling set also defines a sketching method by choosing the $k$-mers from the set as representatives. Although current methods use one of a small number of sketching method families, the space of decycling sets is much larger, and largely unexplored. Finding decycling sets with desirable characteristics (e.g., small remaining path length) is a promising approach to discovering new sketching methods with improved performance (e.g., with small window guarantee). The Minimum Decycling Sets (MDSs) are of particular interest because of their minimum size. Only two algorithms, by Mykkeltveit and Champarnaud, are previously known to generate two particular MDSs, although there are typically a vast number of alternative MDSs. We provide a simple method to enumerate MDSs. This method allows one to explore the space of MDSs and to find MDSs optimized for desirable properties. We give evidence that the Mykkeltveit sets are close to optimal regarding one particular property, the remaining path length. A number of conjectures and computational and theoretical evidence to support them are presented. Code available at https://github.com/Kingsford-Group/mdsscope.