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 → 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 > 0, then we say that f is -far from being submodular. The aim is to have an efficient procedure that, given input f that is -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) ...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: C. Seshadhri, Jan Vondrák
• Keywords: Boolean Hypercube, CORR 2010, Education, Property Testing, Submodular Function