Reconfiguration Problems on Submodular Functions

Abstract: Reconfiguration problems require finding a step-by-step transformation between a pair of feasible solutions for a particular problem. The primary concern in Theoretical Computer Science has been revealing their computational complexity for classical problems. This paper presents an initial study on reconfiguration problems derived from a submodular function, which has more of a flavor of Data Mining. Our submodular reconfiguration problems request to find a solution sequence connecting two input solutions such that each solution has an objective value above a threshold in a submodular function $f: 2^{[n]} \to \mathbb{R}_+$ and is obtained from the previous one by applying a simple transformation rule. We formulate three reconfiguration problems: Monotone Submodular Reconfiguration (MSReco), which applies to influence maximization, and two versions of Unconstrained Submodular Reconfiguration (USReco), which apply to determinantal point processes. Our contributions are summarized as follows: 1. We prove that MSReco and USReco are both $\mathsf{PSPACE}$-complete. 2. We design a $\frac{1}{2}$-approximation algorithm for MSReco and a $\frac{1}{n}$-approximation algorithm for (one version of) USReco. 3. We devise inapproximability results that approximating the optimum value of MSReco within a $(1-\frac{1+\epsilon}{n^2})$-factor is $\mathsf{PSPACE}$-hard, and we cannot find a $(\frac{5}{6}+\epsilon)$-approximation for USReco. 4. We conduct numerical study on the reconfiguration version of influence maximization and determinantal point processes using real-world social network and movie rating data.