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
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.
We use “high probability” to refer to probability >2/3.
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.
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 \(x+\mathbf{e}_{i}+\mathbf{e}_{j} \in\mathcal{D}\).
By that we mean, somewhat different, and not an unknown dwarf.
References
Blais, E., Brody, J., Matulef, K.: Property testing lower bounds via communication complexity. In: Proceedings of the 26th Conference on Computational Complexity (CCC), pp. 210–220 (2011)
Briët, J., Chakraborty, S., García-Soriano, D., Matsliah, A.: Monotonicity testing and shortest-path routing on the cube. Combinatorica 32, 35–53 (2012). Conference version in RANDOM 2010
Batu, T., Dasgupta, S., Kumar, R., Rubinfeld, R.: The complexity of approximating the entropy. SIAM J. Comput. 35(1), 132–150 (2005)
Bhattacharyya, A., Fischer, E., Rubinfeld, R., Valiant, P.: Testing monotonicity of distributions over general partial orders. In: Proceedings of Innovations in Computer Science (ICS), pp. 239–252 (2011)
Balcan, M.-F., Harvey, N.: Learning submodular functions. In: Proceedings of the 43rd Annual Symposium on Theory of Computing (STOC), pp. 793–802 (2011)
Chakrabarty, D., Huang, Z.: Testing coverage functions. In: Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP), pp. 170–181 (2012)
Cheraghchi, M., Klivans, A., Kothari, P., Lee, H.K.: Submodular functions are noise stable. In: Proceedings of the 25th Annual Symposium on Discrete Algorithms (SODA) (2012)
Chakrabarty, D., Seshadhri, C.: Optimal bounds for monotonicity and Lipschitz testing over the hypercube. Technical Report TR12-030, ECCC (2012)
Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A.: Improved testing algorithms for monotonicity. In: Proceedings of the 3rd International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), pp. 97–108 (1999)
Edmonds, J.: Matroids, submodular functions and certain polyhedra. In: Combinatorial Structures and Their Applications, pp. 69–87 (1970)
Fischer, E.: The art of uninformed decisions: a primer to property testing. Bull. Eur. Assoc. Theor. Comput. Sci. 75, 97–126 (2001)
Fischer, E.: On the strength of comparisons in property testing. Inf. Comput. 189(1), 107–116 (2004)
Fischer, E., Lehman, E., Newman, I., Raskhodnikova, S., Rubinfeld, R., Samorodnitsky, A.: Monotonicity testing over general poset domains. In: Proceedings of the 34th Annual Symposium on Theory of Computing (STOC), pp. 474–483 (2002)
Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions ii. Math. Program. Stud. 8, 73–87 (1978)
Fattal, S., Ron, D.: Approximating the distance to monotonicity in high dimensions. http://www.eng.tau.ac.il/~danar/Public-pdf/app-mon-long.pdf
Frank, A.: Matroids and submodular functions. In: Annotated Biblographies in Combinatorial Optimization, pp. 65–80 (1997)
Goldreich, O., Goldwasser, S., Lehman, E., Ron, D., Samordinsky, A.: Testing monotonicity. Combinatorica 20, 301–337 (2000). Conference Version in FOCS 1998
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998). Conference version in FOCS 1996
Goemans, M., Harvey, N., Iwata, S., Mirrokni, V.: Approximating submodular functions everywhere. In: Proceedings of 22th Annual Symposium on Discrete Algorithms (SODA), pp. 535–544 (2009)
Gilbert, E.N.: Gray codes and paths on the n-cube. Bell Syst. Tech. J. 37, 815–826 (1958)
Goldreich, O.: Combinatorial property testing—a survey. In: Randomization Methods in Algorithm Design, pp. 45–60 (1998)
Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. J. ACM 48, 4 (2001). Conference version in STOC 2000
Lovász, L.: Submodular functions and convexity. In: Mathematical Programming: the State of the Art, pp. 235–257 (1983)
Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions i. Math. Program. 14, 265–294 (1978)
Parnas, M., Ron, D., Rubinfeld, R.: On testing convexity and submodularity. SIAM J. Comput. 32(5), 1158–1184 (2003). Conference version in RANDOM 2002
Ron, D.: Property testing. In: Handbook on Randomization, vol. II, pp. 597–649 (2001)
Rubinfeld, R., Sudan, M.: Robust characterization of polynomials with applications to program testing. SIAM J. Comput. 25, 647–668 (1996)
Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory, Ser. B 80, 346–355 (2000)
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 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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DOI: https://doi.org/10.1007/s00453-012-9719-2