Graph decomposition methods compared for query processing speed

Many algorithmic frameworks exist for evaluating join queries. They range from simple ones, such as dynamic programming over a single tree decomposition, high-degree-low-degree splitting from graph algorithms, and the degree uniformization of Joglekar and Ré, to sophisticated ones based on information-theoretic inequalities, such as the PANDA framework for submodular width. Each of these frameworks except high-degree-low-degree splitting comes with a width notion of the query that determines its running-time exponent. We fill this gap by formalizing high-degree-low-degree splitting with $\ell$ degree classes, which gives rise to the degree-splitting width $\mathrm{ds}_\ell$-width. These width notions form a hierarchy, but it was not known whether the more complicated frameworks achieve better worst-case exponents than the simpler ones on any graph query. We study this question for full self-join-free join queries with binary relations, which amounts to colored subgraph isomorphism for a pattern graph $H$. We show that every inequality in the hierarchy $\mathrm{subw} \le \#\mathrm{subw} \le \text{m-width} \le \mathrm{ds}_\ell\text{-width}$ can be strict. For instance, $\text{m-width}(K_{3,3}) = 2 < \mathrm{ds}_\ell\text{-width}(K_{3,3})$ for every constant $\ell$, so no constant number of degree classes reaches the exponent of degree uniformization. We find the smallest separating patterns, with 8, 9, and 6 nodes, respectively, by an exhaustive search with a software tool that computes these widths via mixed integer linear programming. Along the way, we give a degree-based algorithm that removes the polylogarithmic overhead of degree uniformization when evaluating full join queries. Furthermore, we show that with logarithmically many degree classes, the degree-splitting width coincides with the m-width of Joglekar and Ré. Finally, we discuss several open problems.