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Is Submodularity Testable?

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Abstract

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.

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Notes

  1. We deal with one-sided error testers here. If we allow a probability of error in both cases, that would be a two-sided error tester.

  2. We use “high probability” to refer to probability >2/3.

  3. A lattice is any partial order with the operations of “meet” and “join”. In our setting, this means a subset of {0,1}n closed under taking coordinate-wise minimum and maximum. Or equivalently, a family of sets closed under taking intersections and unions.

  4. A set \(\mathcal{D} \subset\{0,1\}^{n}\) is down-closed if \(y \in\mathcal {D}\), x≤y implies \(x \in\mathcal{D}\). We call a function on \(\mathcal{D}\) submodular, if f(x+e i )+f(x+e j )≥f(x)+f(x+e i +e j ) whenever \(x+\mathbf{e}_{i}+\mathbf{e}_{j} \in\mathcal{D}\).

  5. By that we mean, somewhat different, and not an unknown dwarf.

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Acknowledgements

We thank Deeparnab Chakrabarty for useful discussions. Indeed, the main question whether submodularity is testable came up during discussions with him.

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This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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Authors and Affiliations

  1. Sandia National Labs, Livermore, CA, USA

    C. Seshadhri

  2. IBM Almaden Research Center, San Jose, CA, USA

    Jan Vondrák

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  1. C. Seshadhri
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  2. Jan Vondrák
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Correspondence to C. Seshadhri.

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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.

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Seshadhri, C., Vondrák, J. Is Submodularity Testable?. Algorithmica 69, 1–25 (2014). https://doi.org/10.1007/s00453-012-9719-2

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