A Radius-Sensitive Approximation Algorithm for Connected Submodular Maximization

Philip Cervenjak, Junhao Gan, Naonori Kakimura, Seeun William Umboh, Anthony Wirth

Keywords: Combinatorial Optimization, Submodular Maximization, Network Design, Approximation Algorithms, Graph Algorithms

TL;DR: We study the graph problem of Connected Submodular Maximization and the parameter r, the radius of the optimal solution. We achieve a \Omega(\epsilon^3 / r^{\epsilon})-approximation in polynomial time for constant \epsilon in (0, 1].

Abstract:

Connected Submodular Maximization (CSM) is a graph problem with important applications to wireless network deployment, epidemic outbreaks, and cancer genome studies. In CSM, we are given a graph $G$, a non-negative monotone submodular function $f$ on the vertex set of $G$, and an integer $k$. The goal is to select a tree in $G$, with $k$ edges, whose vertex set maximizes $f$. We also study the more general Directed and Directed Rooted variants of CSM (DCSM and DRCSM respectively). In both variants, $G$ is directed and the solution must be an out-tree in $G$, with $k$ edges, whose vertex set maximizes $f$; DRCSM further specifies a vertex to be the root of the selected out-tree. For CSM, several previous works have proposed polynomial time approximation algorithms; the state of the art polynomial time algorithm achieves a $\Omega(\frac{1}{\sqrt{k}})$-approximation. We can also parameterize the approximation factor by the radius of the optimal solution, denoted by $r$; the state of the art polynomial time algorithm achieves a $\Omega(\frac{1}{r})$-approximation.

In this paper, we improve on the state of the art approximation factor for CSM with respect to $r$ as well as $k$, noting that $r \leq k$. We propose a polynomial time framework that, for (Directed) CSM, achieves a $\Omega(\frac{\varepsilon^{3}}{{r}^{\varepsilon}})$-approximation for every constant $\varepsilon \in (0, 1]$. For DRCSM, our framework achieves a $\Omega(\frac{\delta \varepsilon^{3}}{{r}^{\varepsilon}})$-approximation that violates the size constraint by at most a factor of $1 + \delta$ for every $\delta \in [\frac{1}{k}, 1]$. A key component of our framework is GreedyRadius, an algorithm for DRCSM that outputs a bicriteria approximation, i.e., an approximate solution that violates the size constraint by at most some factor. GreedyRadius takes an algorithm with a bicriteria approximation factor in terms of $k$ and outputs a solution with the same bicriteria approximation factor (up to constants) in terms of $r$. Moreover, to use as a subroutine for DRCSM, we propose the algorithm RecApprox-$d$, which achieves a $\frac{1}{d+1}$-approximation that violates the size constraint by at most a factor of $(d+1)^{2} k^{\frac{1}{d}}$. RecApprox-$d$ uses a recursive greedy strategy, with $d$ denoting the number of levels of recursion used. This enables the dependence on $\varepsilon$ in the approximation factors of our overall framework.

Area: Search, Optimization, Planning, and Scheduling (SOPS)

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Submission Number: 947