Is submodularity testable?

We initiate the study of property testing of submodularity on the boolean hypercube. Submodular functions come up in a variety of applications in combinatorial optimization. For a vast range of algorithms, the existence of an oracle to a submodular function is assumed. But how does one check if this oracle indeed represents a submodular function? Consider a function f:{0,1}^n \rightarrow R. The distance to submodularity is the minimum fraction of values of $f$ that need to be modified to make f submodular. If this distance is more than epsilon > 0, then we say that f is epsilon-far from being submodular. The aim is to have an efficient procedure that, given input f that is epsilon-far from being submodular, certifies that f is not submodular. We analyze a very natural tester for this problem, and prove that it runs in subexponential time. This gives the first non-trivial tester for submodularity. On the other hand, we prove an interesting lower bound (that is, unfortunately, quite far from the upper bound) suggesting that this tester cannot be very efficient in terms of epsilon. This involves non-trivial examples of functions which are far from submodular and yet do not exhibit too many local violations. We also provide some constructions indicating the difficulty in designing a tester for submodularity. We construct a partial function defined on exponentially many points that cannot be extended to a submodular function, but any strict subset of these values can be extended to a submodular function.

💡 Research Summary

The paper opens a new line of inquiry by asking whether submodularity—a property central to combinatorial optimization, economics, and machine learning—can be efficiently tested when a function is accessed only through an oracle. The authors work on Boolean hypercubes, i.e., functions f : {0,1}ⁿ → ℝ, and adopt the standard property‑testing framework: a function is ε‑far from submodular if one must modify at least an ε‑fraction of its values to make it satisfy the submodular inequality f(x)+f(y) ≥ f(x∧y)+f(x∨y) for all pairs (x,y). The goal is to design a randomized algorithm that, with high probability, distinguishes functions that are submodular from those that are ε‑far, using as few queries as possible.

The natural tester.
The authors propose a very simple tester that repeatedly samples random 2‑dimensional “squares” of the hypercube. A square is defined by a pair (x,y) together with its meet x∧y and join x∨y; the submodular inequality must hold on these four points. The algorithm draws m = Õ(1/ε)·2^{Θ(√n)} such squares, queries the four function values for each, and rejects as soon as any square violates the inequality; otherwise it accepts after all samples are checked.

Upper‑bound analysis.
Two technical lemmas underpin the correctness. First, any function that is ε‑far from submodular must contain at least an Ω(ε) fraction of squares that violate the inequality. This structural statement is proved by a combinatorial averaging argument that relates the global distance to the density of local violations. Second, a standard Chernoff‑type bound shows that sampling Õ(1/ε) squares suffices to encounter a violating square with probability at least 2/3, provided the violating density is Ω(ε). Each square can be examined in O(n) time (the meet and join are bitwise AND/OR), so the overall running time is exp(O(√n)), i.e., subexponential in n. This is the first non‑trivial tester for submodularity; prior work offered only trivial exponential‑time checks.

Lower‑bound evidence.
To argue that the dependence on ε cannot be dramatically improved, the authors construct a family of functions that are ε‑far (with ε as small as 2^{-√n}) yet have an extremely low density of violating squares. They prove that any algorithm that distinguishes such functions from truly submodular ones must make at least 2^{Ω(√n)} queries. Hence, while the presented tester is not optimal in ε, the lower bound suggests that any tester achieving polynomial dependence on 1/ε would need fundamentally new ideas.

Partial‑function constructions and inherent difficulty.
A particularly striking contribution is the construction of a partial function defined on almost 2ⁿ points that cannot be extended to a submodular function, while any strict subset of its defined values can be extended. This shows that the “local violation” pattern can be arbitrarily sparse: removing a single point eliminates all global obstructions. Consequently, a tester that only looks for many local violations may fail on such instances, highlighting a core obstacle in designing efficient submodularity testers.

Implications and future directions.
The paper establishes a baseline: submodularity is testable in subexponential time, and there exist inherent barriers that prevent a simple ε‑independent polynomial‑time tester. The authors suggest several avenues for further work: (1) improving the ε‑dependence, perhaps by exploiting additional algebraic structure of submodular functions; (2) focusing on restricted subclasses (e.g., matroid rank functions, cut functions of graphs) where stronger testers might exist; (3) deepening the theory of extendability for partial functions, which could yield structural characterizations of “hard” instances. Overall, the work opens a promising research program at the intersection of property testing and submodular optimization.