A Non-Extendibility Certificate for Submodularity and Applications

Umang Bhaskar, Gunjan Kumar

Abstract: Can a function f defined on some domain $$\mathcal {D}$$ be extended to a submodular function on a larger domain $$\mathcal {D}' \supset \mathcal {D}$$ ? This is the problem of submodular partial function extension. In this work, we develop a new combinatorial certificate of nonextendibility called a square certificate. We then present two applications of our certificate: to submodular extension on lattices, and to property testing of submodularity. - For lattices, we define a new class of lattices called pseudocyclic lattices that strictly generalize modular lattices, and show that these are sublattice extendible, i.e., a partial function that is submodular on a sublattice is extendible to a submodular function on the lattice. We give an example to show that in general lattices this property does not hold. - For property testing, we show general lower bounds for a class of submodularity testers called proximity oblivious testers. One of our lower bounds is applicable to matroid rank functions as well, and is the first lower bound for this class of functions.