[Paper Review] Small Space Stream Summary for Matroid Center

Sagar Kale|arXiv (Cornell University)|Oct 15, 2018

Facility Location and Emergency ManagementBusiness, Management and Accounting26 references3 citations

TL;DR

This paper presents the first one-pass streaming algorithms for the matroid center problem and its outlier variant, achieving (7+ε)-approximation with O(r² log(1/ε)/ε) space and (15+ε)-approximation for the outlier version. It establishes a Ω(r²) space lower bound for better-than-∆-approximation, revealing a quadratic gap between matroid center and k-center in streaming, and provides efficient two-pass algorithms with improved approximation ratios using stream summarization and stream-strapping techniques.

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 $Δ$-approximation for partition-matroid center must use $Ω(r^2)$ bits of space, where $Δ$ 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 $O((nr + r^{3.5})\log(1/\varepsilon)/\varepsilon + r^2(\log Δ)/\varepsilon)$, 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).

Motivation & Objective

• To design efficient one-pass and two-pass streaming algorithms for the matroid center problem with sublinear space.
• To establish a strong space complexity lower bound for randomized one-pass streaming algorithms for matroid center.
• To extend the framework to matroid center with z outliers and knapsack center variants.
• To achieve approximation ratios close to the known offline best (3+ε) using minimal space and efficient computation.
• To demonstrate that matroid constraints fundamentally increase space requirements compared to k-center in the streaming model.

Proposed method

• Uses a stream-strapping technique to reduce space usage by maintaining multiple candidate solutions with increasing thresholds.
• Maintains a stream summary of O(r² log(1/ε)/ε) points that contains a (7+ε)-approximate solution.
• Employs a brute-force search on the summary for (7+ε)-approximation or an efficient algorithm for (17+ε)-approximation.
• Applies a two-pass approach to compute a (3+ε)-approximation efficiently, leveraging the summary and matroid independence checks.
• Extends techniques to knapsack center and knapsack center with outliers, achieving space linear in r (not r²), improving over matroid center.
• Uses a doubling-style thresholding strategy with active and inactive copies to handle unknown optimal cost.

Experimental results

Research questions

• RQ1 What is the minimum space required for a one-pass randomized streaming algorithm to achieve better than ∆-approximation in matroid center?
• RQ2 Can we design a one-pass streaming algorithm for matroid center that achieves a (7+ε)-approximation with space subquadratic in r?
• RQ3 How does the space complexity of matroid center compare to k-center in the streaming model?
• RQ4 Can we extend the framework to matroid center with z outliers and knapsack center with similar approximation guarantees and space efficiency?
• RQ5 What is the best approximation ratio achievable in two passes with subquadratic space for matroid center?

Key findings

• Any randomized one-pass streaming algorithm for matroid center that achieves better than ∆-approximation must use Ω(r²) bits of space, establishing a quadratic separation from k-center.
• A one-pass algorithm achieves a (7+ε)-approximation using O(r² log(1/ε)/ε) space, with a brute-force search on the summary.
• An efficient one-pass algorithm achieves a (17+ε)-approximation using the same space, avoiding brute-force search.
• A two-pass algorithm computes a (3+ε)-approximation in O((nr + r³.⁵)log(1/ε)/ε + r² log∆/ε) total time and O(r² log(1/ε)/ε) space.
• For matroid center with z outliers, a one-pass algorithm stores O((r² + rz)log(1/ε)/ε) points and achieves a (15+ε)-approximation with brute force or (51+ε)-approximation with efficient computation.
• The framework extends to knapsack center, reducing space complexity from O(r²) to O(r) for the same approximation ratios.