Fast Hypertree Decompositions via Linear Programming: Fractional and Generalized

Efficient query evaluation in databases and solving constraint satisfaction problems (CSPs) are crucial for improving performance in many real-world applications, from large-scale data management to decision-making systems. These problems naturally admit hypergraph representations, and are efficiently solvable using hypertree decomposition techniques, when the decomposition width is small. However, these techniques require finding small-width decompositions efficiently. This remains a significant challenge despite research from both the database and theory communities. In this work we present Ralph (Randomized Approximation using Linear Programming for Hypertree-Decompositions), a fast algorithm to compute low width fractional and generalized hypertree decompositions for input hypergraphs, as well as lower bounds for these widths. We build on the recent breakthrough by Korchemna et al. [FOCS 2024], which introduced the first polynomial time approximation algorithm for fractional (generalized) hypertree width. Our approach combines this theoretical advancement with practical heuristic improvements utilizing (mixed-integer) linear programs. Along the way, we present new algorithms with strong theoretical guarantees. Through empirical evaluation on the nearly 3700 instances of HyperBench, a well established benchmark suite for hypertree decompositions, we find near optimal decompositions for all previously solved instances and low width decompositions for all 500 previously unsolved instances, effectively pushing state-of-the-art.