ICS 2011

Welcome to ICS2011
Innovations in Computer Science  ICS 2011, Tsinghua University, Beijing, China, January 79, 2011. Proceedings, 195210, 9787302245179
Tsinghua University Press 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 εfar from being submodular. The aim is to have an efficient procedure that, given input f that is εfar 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 nontrivial 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 ε. This involves nontrivial examples of functions which are far from being 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. Preview:

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