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A famous example of the periodicity forcing word equation is $xy = yx$. It means that if a word $xy$ equals to a word $yx$, obtained by a _cut-and-swap_ process, there exists a word $w$ and integers $i,j$ such that $x = w^i$ and $y = w^j$. The paper's Section 1 and 5 contain a brief yet very helpful overview of existing results about periodicity forcing. About ten years ago, Czeizler, Kari and Seki, in their work [On a special class of primitive words](https://www.csd.uwo.ca/~lkari/mfcs.pdf), have introduced a notion of a pseudo-repetition. It can be explained with the following example. When DNA molecules are regarded as words, the Watson-Crick correspondence $$G \sim C, A \sim T$$ between guanine ($G$) and cytosine ($C$), adenine ($A$) and thymine ($T$) allows us to say that a word $ATCGATTGAGCTCTAGCG$ contains the same information as the word $TAGCTAACTCGAGATCGC$. A pseudo-repetition is a repetition of a word or its Watson-Crick complement. So, for instance $ATC \sim TAG$, and the word $ATCTAGATCATCTAG$ is a pseudo-repetition of the word $ATC$. In an abstract algebraic setting (involving monoids, morphic permutations, equivalence classes and anticongruences), the author studies an extension to the notion of periodicity forcing, using the recently introduced concept of pseudo-repetitions and related ideas. Holub proves, for instance, that any periodicity forcing classical word equation also forces a pseudo-periodicity. He provides interesting new results and generalizations of already known results. MathSciNet:MR4074724.
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Nice coarse-gaining based heuristic to partition the nodes into k sets of similar sizes minimizing the number of edges between those sets. Random maximal matchings are used to coarse-grain the graph. Maybe fast heuristics to obtain a large maximal matching can be considered, such as using the two rules detailed here: [Karp-Sipser based kernels for bipartite graph matching](https://papers-gamma.link/paper/160). Publicly available implementation: http://glaros.dtc.umn.edu/gkhome/metis/metis/overview
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How to efficiently generate combinatorial structures such as posets, trees, B-trees, graphs, permutations, set partitions, Dyck paths, multisets or necklaces ... ? The Ruskey's book gives answers. It contains many surprising descriptions (and bibliographic references) of constant amortized time (CAT) and loopless algorithms.   > **CAT Algorithms** The holy grail of generating combinatorial objects is to find an algorithm that runs in Constant Amortized Time. This means that the amount of computation, after a small amount of preprocessing, is proportional to the number of objects that are listed. We do not count the time to actually output or process the objects; we are only concerned with the amount of data structure change that occurs as the objects are being generated.     _ Ruskey's book, page 8 _   > ** Loopless algorithm ** is an imperative algorithm that generates successive combinatorial objects, such as partitions, permutations, and combinations, in constant time and the first object in linear time     _ https://en.wikipedia.org/wiki/Loopless_algorithm _
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