Maximizing stochastic set function under a matroid constraint from decomposition

In this work, we focus on maximizing the stochastic DS decomposition problem. If the constraint is a uniform matroid, we design an adaptive policy, namely Myopic Parameter Conditioned Greedy, and prove its theoretical guarantee \(f(\varTheta (\pi _k))-(1-c_G)g(\varTheta (\pi _k))\ge (1-e^{-1})F(\pi ^*_A, \varTheta (\pi _k)) - G(\pi ^*_A,\varTheta (\pi _k))\), where \(F(\pi ^*_A, \varTheta (\pi _k)) = \mathbb {E}_{\varTheta }[f(\varTheta (\pi ^*_A)) \vert \varTheta (\pi _k)]\). When the constraint is a general matroid constraint, we design the Parameter Measured Continuous Conditioned Greedy to return a fractional solution. To round an integer solution from the fractional solution, we adopt the lattice contention resolution and prove that there is a \((b, \frac{1-e^{-b}}{b})\) lattice CR scheme under a matroid constraint. Additionally, we adopt the pipage rounding to obtain a non-adaptive policy with the theoretical guarantee \(F(\pi )-(1-c_G)G(\pi ) \ge (1-e^{-1}) F(\pi ^*_A) - G(\pi ^*_A) - O(\epsilon )\) and utlize the \((1,1-e^{-1})\)-lattice contention resolution scheme \(\tau \) to obtain an adaptive solution \(\mathbb {E}_{\tau \sim \varLambda } [f(\tau (\varTheta (\pi )))- (1-c_G) g(\tau (\varTheta (\pi )))] \ge (1-e^{-1})^2F(\pi ^*_A,\varTheta (\pi )) - (1-e^{-1}) G(\pi ^*_A,\varTheta (\pi )) -O(\epsilon )\). Since any set function can be expressed as the DS decomposition, our framework provides a method for solving the maximization problem of set functions defined on a random variable set.