With the extensive application of submodularity, its generalizations are constantly being proposed. However, most of them are tailored for special problems. In this paper, we focus on quasi-submodularity, a universal generalization, which satisfies weaker properties than submodularity but still enjoys favorable performance in optimization. Similar to the diminishing return property of submodularity, we first define a corresponding property called the single sub-crossing , then we propose two algorithms for unconstrained quasi-submodular function minimization and maximization, respectively. The proposed algorithms return the reduced lattices in @math iterations, and guarantee the objective function values are strictly monotonically increased or decreased after each iteration. Moreover, any local and global optima are definitely contained in the reduced lattices. Experimental results verify the effectiveness and efficiency of the proposed algorithms on lattice reduction.
Quasi-supermodularity stems from economic fields. Milgrom and Shannon @cite_10 first propose the definition of quasi-supermodularity. They find that the maximizer of a quasi-supermodular function is monotone as the parameter changes. In combinatorial optimization, for quasi-submodular functions, this property means the set of minimizers has a nested structure, which is the foundation of the proposed UQSFMin algorithm. Based on the theorem above, suppose we start from @math , if @math , @math , then we can set @math . This theorem ensures that there exists a chain structure of minimizers. This is a general principle. First, it works in submodular cases, for submodularity is a strict subset of quasi-submodularity. Moreover, when the superdifferential in @cite_21 is not superdifferential for non-submodular quasi-submodular functions, such as the determinant function and the multiplicatively separable functions, this principle can also hold. In @cite_10 , only quasi-submodular function minimization (or equivalently, quasi-supermodular function maximization) is considered. For quasi-submodular function maximization, there is no existing study.
@cite_21 @cite_10
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A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits a certain "symmetry gap" in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1-1 e)-approximation for monotone submodular maximization under a cardinality constraint, and the impossibility of (1 2+epsilon)-approximation for unconstrained (non-monotone) submodular maximization. It follows from our result that (1 2+epsilon)-approximation is also impossible for non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309-approximation for this problem, improving the previously known factor of 1 4-o(1).As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A (1 6-o(1))-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k≫=2, there is a class of matroids of fractional base packing number nu = k (k-1), such that any algorithm achieving a better than (1-1 nu)-approximation for this class would require exponentially many value queries. On the positive side, we present a 1 2 (1-1 nu-o(1))-approximation algorithm for the same problem. Our hardness results hold in fact for very special symmetric instances. For such symmetric instances, we show that the approximation factors of 1 2 (for submodular maximization subject to a matroid constraint) and 1-1 nu (for a matroid base constraint) can be achieved algorithmically and hence are optimal. Submodularity is one of the most important properties of combinatorial optimization, and k-submodularity is a generalization of submodularity. Maximization of a k-submodular function is NP-hard, and approximation algorithms have been studied. Most of algorithms use randomization and achieve the approximation ratio as the expected value. For unconstrained submodular maximization, [Buchbinder and Feldman 2016] gave a derandomization scheme, and showed that randomness is not necessary for obtaining optimal results. In this paper, we extend their scheme for unconstrained k-submodular maximization and give two deterministic algorithms for k-submodular maximization. The first is for monotone k-submodular functions. We derandomize (k (2k-1) )-approximation algorithm [Iwata, Tanigawa, and Yoshida 2016], and achieve the same approximation ratio. The second is for nonmonotone functions with (k 2 ). It achieves (k (3k-2) )-approximation, and it is better than a simple greedy algorithm [Ward and Živný 2016] which achieves 1 3-approximation for (k 3 ). Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. We consider the problem of maximizing a nonnegative (possibly non-monotone) submodular set function with or without constraints. [9] showed a 2 5-approximation for the unconstrained problem and also proved that no approximation better than 1 2 is possible in the value oracle model. Constant-factor approximation has been also known for submodular maximization subject to a matroid independence constraint (a factor of 0.309 [33]) and for submodular maximization subject to a matroid base constraint, provided that the fractional base packing number ν is bounded away from 1 (a 1 4-approximation assuming that ν ≥ 2 [33]). In this paper, we propose a new algorithm for submodular maximization which is based on the idea of simulated annealing. We prove that this algorithm achieves improved approximation for two problems: a 0.41-approximation for unconstrained submodular maximization, and a 0.325-approximation for submodular maximization subject to a matroid independence constraint. On the hardness side, we show that in the value oracle model it is impossible to achieve a 0.478-approximation for submodular maximization subject to a matroid independence constraint, or a 0.394-approximation subject to a matroid base constraint in matroids with two disjoint bases. Even for the special case of cardinality constraint, we prove it is impossible to achieve a 0.491-approximation. (Previously it was conceivable that a 1 2-approximation exists for these problems.) It is still an open question whether a 1 2-approximation is possible for unconstrained submodular maximization.