Cuts and Gauges for Submodular Width
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AIPR assessment
The problem is hard and competitive, since submodular width and its relationship to ghw sit in a mature line of work where progress has been incremental and technically difficult. The strengths reinforce each other well: a clean new reformulation, a solid proof architecture, and concrete structural consequences for query evaluation. The main weakness is that the practical payoff remains indirect, because the results are a theory framework rather than an implementable algorithm or evaluated syste
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 \[ subw(H) \in Ω\left(\frac{ghw(H)}{\log ghw(H)} \right). \]
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