Small Space Stream Summary for Matroid Center

TL;DR

This paper investigates streaming algorithms for the matroid center problem, establishing space complexity bounds and providing approximation algorithms with various passes and extensions to related problems.

Contribution

It introduces new space lower bounds and efficient approximation algorithms for matroid center in streaming settings, including extensions to outliers and knapsack variants.

Findings

Any better than $ ext{Delta}$-approximation requires $ ext{Omega}(r^2)$ space.

A one-pass stream summary algorithm achieves a $(7+ ext{epsilon})$-approximation.

With two passes, a $(3+ ext{epsilon})$-approximation is efficiently computed.

Abstract

In the matroid center problem, which generalizes the $k$-center problem, we need to pick a set of centers that is an independent set of a matroid with rank $r$. We study this problem in streaming, where elements of the ground set arrive in the stream. We first show that any randomized one-pass streaming algorithm that computes a better than $\Delta$-approximation for partition-matroid center must use $\Omega(r^2)$ bits of space, where $\Delta$ is the aspect ratio of the metric and can be arbitrarily large. This shows a quadratic separation between matroid center and $k$-center, for which the Doubling algorithm gives an $8$-approximation using $O(k)$-space and one pass. To complement this, we give a one-pass algorithm for matroid center that stores at most $O(r^2\log(1/\varepsilon)/\varepsilon)$ points (viz., stream summary) among which a $(7+\varepsilon)$-approximate solution exists, which can be found by brute force, or a $(17+\varepsilon)$-approximation can be found with an efficient algorithm. If we are allowed a second pass, we can compute a $(3+\varepsilon)$-approximation efficiently; this also achieves almost the known-best approximation ratio (of $3+\varepsilon$) with total running time of $O((nr + r^{3.5})\log(1/\varepsilon)/\varepsilon + r^2(\log \Delta)/\varepsilon)$, where $n$ is the number of input points. We also consider the problem of matroid center with $z$ outliers and give a one-pass algorithm that outputs a set of $O((r^2+rz)\log(1/\varepsilon)/\varepsilon)$ points that contains a $(15+\varepsilon)$-approximate solution. Our techniques extend to knapsack center and knapsack center with outliers in a straightforward way, and we get algorithms that use space linear in the size of a largest feasible set (as opposed to quadratic space for matroid center).