Cuts and Gauges for Submodular Width

2026-04-24

AI Summary

This work uncovers the geometric essence of submodular width, a central measure in the complexity of conjunctive query evaluation. By introducing a novel branchwidth based on hypergraph edge cuts and costs induced by admissible submodular functions, the paper provides the first variational characterization of submodular width from a convex-geometric perspective and establishes connections to classical parameters such as line graph treewidth and multicommodity flow. The main contributions include a new branchwidth that admits an efficient 3/2-approximation and a proof that, under natural conditions, submodular width satisfies subw(H) ∈ Ω(ghw(H)/log ghw(H)), yielding an asymptotic lower bound relative to generalized hypertree width. The approach integrates hypergraph cut analysis, submodular optimization, and graph decomposition theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Submodular width is a central structural measure governing the complexity of conjunctive query evaluation. In this paper we recast submodular width in geometric terms. We how that submodular width can be approximated, up to a factor $3/2$, by a new branchwidth parameter defined in terms of edge separations in the hypergraph and the costs induced on them by admissible submodular functions. This reformulation turns lower bounds on submodular width into the problem of constructing well-balanced edge separations whose induced cost remains small. We then express this connection through a variational characterisation in terms of a convex body. Using these tools, we relate submodular width to more familiar graph-theoretic notions, including line-graph treewidth and multicommodity flow, and obtain general conditions under which submodular width is tightly linked to generalised hypertree width. In particular, under various natural conditions we show that \[ subw(H) \in Ω\left(\frac{ghw(H)}{\log ghw(H)} \right). \]

Problem

• submodular width
• conjunctive query evaluation
• hypergraph
• generalised hypertree width
• branchwidth

Innovation

• submodular width
• branchwidth
• edge separations
• convex body
• generalised hypertree width