Reconfiguration Problems on Submodular Functions

Naoto Ohsaka, Tatsuya Matsuoka

Abstract: emphReconfiguration 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 $f: 2^[n] \to \mathbbR _+$ 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: \beginitemize \item We prove that MSReco and USReco are both PSPACE-complete. \item We design a $\frac1 2 $-approximation algorithm for MSReco and a $\frac1 n $-approximation algorithm for (one version of) USReco. \item We devise inapproximability results that approximating the optimum value of MSReco within a $(1-\frac1+ε n^2 )$-factor is PSPACE-hard, and we cannot find a $(\frac5 6 +ε)$-approximation for USReco. \item We conduct numerical study on the reconfiguration version of influence maximization and determinantal point processes using real-world social network and movie rating data. \enditemize