, Volume 69, Issue 1, pp 1-25

Is Submodularity Testable?

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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 →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 ϵ>0, then we say that f is ϵ-far from being submodular. The aim is to have an efficient procedure that, given input f that is ϵ-far from being submodular, certifies that f is not submodular. We analyze a 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 efficient in terms of ϵ. 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.

This work was funded by the applied mathematics program at the United States Department of Energy and performed at Sandia National Laboratories, a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.