Cuts and Gauges for Submodular Width

TL;DR

This paper introduces a geometric reformulation of submodular width, connecting it to graph-theoretic concepts and providing approximation and lower bound techniques for query evaluation complexity.

Contribution

It recasts submodular width in geometric terms, enabling approximation, bounds, and connections to known graph parameters like treewidth and hypertree width.

Findings

Submodular width can be approximated within a factor of 3/2 using a new branchwidth parameter.

The paper relates submodular width to line-graph treewidth and multicommodity flow.

Under natural conditions, submodular width is tightly linked to generalized hypertree width, with bounds involving logarithmic factors.

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 \[ subw(H) \in \Omega \left(\frac{ghw(H)}{\log ghw(H)} \right). \]