We prove that any submodular function f: 0,1 ^n -> 0,1,...,k can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership queries under the uniform distribution for submodular functions of the form f: 0,1 ^n -> 0,1,...,k . Our algorithm runs in time polynomial in n, k^ O(k k ) , 1 and log(1 ) and works even in the agnostic setting. The line of previous work on learning submodular functions [Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi, Klivans, Kothari, Lee (SODA '12)] implies only n^ O(k) query complexity for learning submodular functions in this setting, for fixed epsilon and delta. Our learning algorithm implies a property tester for submodularity of functions f: 0,1 ^n -> 0, ..., k with query complexity polynomial in n for k=O(( n n)^ 1 2 ) and constant proximity parameter .
For the special case of Boolean functions, characterizations of submodular and monotone submodular functions in terms of simple DNF formulas are known. A Boolean function is monotone submodular if and only if it can be represented as a monotone 1-DNF (see, e.g., Appendix A in @cite_25 ). A Boolean function is submodular if and only if it has a pure (without singleton terms) 2-DNF representation @cite_7 .
@cite_25 @cite_7
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We prove that any submodular function f: 0, 1 n → 0, 1,..., k can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership queries under the uniform distribution for submodular functions of the form f: 0, 1 n → 0, 1,..., k . Our algorithm runs in time polynomial in n, Abstract After providing a simple characterization of Horn functions (i.e., those Boolean functions that have a Horn DNF), we study in detail the special class of submodular functions. Every prime implicant of such a function involves at most one complemented and at most one uncomplemented variable, and based on this we give a one-to-one correspondence between submodular functions and partial preorders (reflexive and transitive binary relations), and in particular between the nondegenerate acyclic submodular functions and the partially ordered sets. There is a one-to-one correspondence between the roots of a submodular function and the ideals of the associated partial preorder. There is also a one-to-one correspondence between the prime implicants of the dual of the submodular function and the maximal antichains of the associated partial preorder. Based on these results, we give graph-theoretic characterizations for all minimum prime DNF representations of a submodular function. The problem of recognizing submodular functions in DNF representation is coNP-complete. A Boolean function @math is said to be @math -far from monotone if @math needs to be modified in at least @math -fraction of the points to make it monotone. We design a randomized tester that is given oracle access to @math and an input parameter @math and has the following guarantee: It outputs Yes if the function is monotonically nondecreasing and outputs No with probability @math , if the function is @math -far from monotone. This nonadaptive, one-sided tester makes @math queries to the oracle. We consider the problem of testing whether an unknown Boolean function f : -- 1, 1 n a#x21C6; -- 1, 1 is monotone versus aepsi;-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied problem. Lower bound: We prove an a#x03A9;(n1 5) lower bound on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotone. This gives an exponential improvement on the previous lower bound of a#x03A9;(log n) due to [1]. We show that the same lower bound holds for monotonicity testing of Boolean-valued functions over hypergrid domains 1,,m n for all m age; 2. Upper bound: We present an O(n5 6) poly(1 aepsi;)-query algorithm that tests whether an unknown Boolean function f is monotone versus aepsi;-far from monotone. Our algorithm, which is non-adaptive and makes one-sided error, is a modified version of the algorithm of Chakrabarty and Seshadhri[2], which makes O(n7 8) poly(1 aepsi;) queries.