Canonical functions: a proof via topological dynamics

Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity

The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on finite domains).Comment: Full version of an MFCS'20 pape

The Complexity of Phylogeny Constraint Satisfaction

We systematically study the computational complexity of a broad class of computational problemsin phylogenetic reconstruction. The class contains for example the rooted triple consistencyproblem, forbidden subtree problems, the quartet consistency problem, and many other problemsstudied in the bioinformatics literature. The studied problems can be described as constraintsatisfaction problems where the constraints have a first-order definition over the rooted triplerelation. We show that every such phylogeny problem can be solved in polynomial time or isNP-complete. On the algorithmic side, we generalize a well-known polynomial-time algorithmof Aho, Sagiv, Szymanski, and Ullman for the rooted triple consistency problem. Our algorithmrepeatedly solves linear equation systems to construct a solution in polynomial time. We thenshow that every phylogeny problem that cannot be solved by our algorithm is NP-complete. Ourclassification establishes a dichotomy for a large class of infinite structures that we believe is ofindependent interest in universal algebra, model theory, and topology. The proof of our mainresult combines results and techniques from various research areas: a recent classification of themodel-complete cores of the reducts of the homogeneous binary branching C-relation, Leeb’sRamsey theorem for rooted trees, and universal algebra