The Expressive Power of Binary Submodular Functions

It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular functions of arity 4 can be expressed by binary submodular functions. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish for the first time that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.

💡 Research Summary

The paper tackles a long‑standing open question in the theory of Boolean submodular functions: can every Boolean submodular function be expressed as a sum of binary (i.e., quadratic) submodular functions, possibly after introducing auxiliary variables? While many special subclasses (negative‑positive polynomials, cubic submodular polynomials, 2‑monotone functions, etc.) are known to be “expressible” in this way, no general technique existed to answer the question for all submodular functions.

The authors adopt a recent algebraic framework for valued constraint satisfaction problems (VCSPs). Central to this framework is the notion of a multimorphism: a tuple of operations that, when applied component‑wise to any collection of feasible assignments, never increases the total cost. For submodular functions on a lattice, the pair (min, max) forms a multimorphism; consequently, any function that admits (min, max) as a multimorphism is submodular. Moreover, multimorphisms are preserved under expressibility: if a set Γ of cost functions has a multimorphism F, then every function expressible over Γ must also admit F. This yields a powerful tool for proving non‑expressibility: exhibit a multimorphism of the target class (binary submodular functions) that is violated by a candidate function.

Using this tool, the authors focus on Boolean submodular polynomials of arity four (quartic). They introduce two families of functions, called upper fans and lower fans, originally defined in earlier work on the cone of submodular functions. An upper fan is determined by a collection of comparable lattice elements that share a common least upper bound; a lower fan is defined dually with a common greatest lower bound. Both families are submodular and, crucially, they admit the (min, max) multimorphism. The authors prove that every upper or lower fan can be expressed as a sum of binary submodular functions using only a linear number of auxiliary variables. This yields a new infinite class of submodular functions of arbitrary arity that are efficiently reducible to a Min‑Cut problem.

The main negative result concerns quartic submodular polynomials that are not fans. By constructing explicit examples and analysing their second‑order discrete derivatives, the authors show that such functions violate a multimorphism that all binary submodular functions possess. Consequently, no gadget consisting of binary submodular functions (even with any number of extra variables) can reproduce the original cost function. Hence, not all Boolean submodular functions are expressible by binary ones.

A striking corollary follows: the well‑known technique of reducing a submodular optimization problem to a Min‑Cut instance (graph cuts) cannot be applied universally. While quadratic submodular functions and the newly identified fan families admit such a reduction, the non‑fan quartic examples do not, establishing a concrete limitation of graph‑cut based algorithms for general submodular minimisation.

Beyond algorithmic implications, the paper settles a conjecture of Promislow and Young concerning the extreme rays of the cone of Boolean submodular functions. The conjecture claimed that all extreme rays correspond to fan‑type functions. The existence of non‑fan quartic submodular functions that are extreme rays disproves this claim. The authors propose a refined conjecture suggesting that extreme rays consist of a mixture of fan and non‑fan structures, opening a new line of investigation into the geometry of the submodular cone.

The paper also discusses practical ramifications for two major application domains. In artificial intelligence, valued CSPs with submodular cost functions are known to be tractable via general submodular minimisation algorithms. However, the authors point out that only those VCSP instances whose constraints are fans can be solved by the more efficient graph‑cut reduction; others must rely on generic SFM algorithms. In computer vision, many energy minimisation problems are formulated as graph‑cut problems. The authors’ characterisation precisely delineates which 4‑ary energy terms can be handled by graph cuts and which cannot, guiding practitioners in model design.

In summary, the paper delivers three core contributions: (1) a complete characterisation of which quartic Boolean submodular functions are expressible by binary submodular functions, (2) the identification of a broad, arbitrarily‑high‑arity class of “fan” functions that are expressible and thus efficiently minimisable via Min‑Cut, and (3) a definitive negative answer to the universal expressibility question, together with the refutation of an existing conjecture about the submodular cone. The work combines algebraic theory (multimorphisms), combinatorial optimisation (graph cuts), and complexity analysis to deepen our understanding of submodular function expressibility and its algorithmic consequences.