On Unconstrained Quasi-submodular Function Optimization

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 satis?es weaker properties than submodularity but still enjoys favorable performance in optimization.Similar to the diminishing return property of submodularity, we ?rst de?ne 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 O(n) iterations, and guarantee the objective function values are strictly monotonically increased or decreased after each iteration. Moreover, any local and global optima are de?nitely contained in the reduced lattices. Experimental results verify the effectiveness and e?ciency of the proposed algorithms on lattice reduction.