Parametric monotone function maximization with matroid constraints

Abstract

We study the problem of maximizing an increasing function $f:2^N\rightarrow \mathcal {R}_{+}$ subject to matroid constraints. Gruia Calinescu, Chandra Chekuri, Martin Pál and Jan Vondrák have shown that, if f is nondecreasing and submodular, the continuous greedy algorithm and pipage rounding technique can be combined to find a solution with value at least $1-1/e$ of the optimal value. But pipage rounding technique have strong requirement for submodularity. Chandra Chekuri, Jan Vondrák and Rico Zenklusen proposed a rounding technique called contention resolution schemes. They showed that if f is submodular, the objective value of the integral solution rounding by the contention resolution schemes is at least $1-1/e$ times of the value of the fractional solution. Let $f:2^N\rightarrow \mathcal {R}_{+}$ be an increasing function with generic submodularity ratio $\gamma \in (0,1]$, and let $(N,\mathcal {I})$ be a matroid. In this paper, we consider the problem $\max _{S\in \mathcal {I}}f(S)$ and provide a $\gamma (1-e^{-1})(1-e^{-\gamma }-o(1))$-approximation algorithm. Our main tools are the continuous greedy algorithm and contention resolution schemes which are the first time applied to nonsubmodular functions.