[Paper Review] Almost Optimal Streaming Algorithms for Coverage Problems

MohammadHossein Bateni, Hossein Esfandiari+1 more|arXiv (Cornell University)|Oct 25, 2016

Complexity and Algorithms in Graphs26 references4 citations

TL;DR

This paper presents almost optimal streaming algorithms for maximum coverage and minimum set cover problems in the edge-arrival model, using a novel sketching technique that enables (1−ε)α-approximation with Õ(n) space, independent of set or element sizes. The approach improves upon prior work by eliminating oracle access assumptions and achieving near-optimal approximation and space complexity in a more realistic streaming setting.

ABSTRACT

Maximum coverage and minimum set cover problems --collectively called coverage problems-- have been studied extensively in streaming models. However, previous research not only achieve sub-optimal approximation factors and space complexities, but also study a restricted set arrival model which makes an explicit or implicit assumption on oracle access to the sets, ignoring the complexity of reading and storing the whole set at once. In this paper, we address the above shortcomings, and present algorithms with improved approximation factor and improved space complexity, and prove that our results are almost tight. Moreover, unlike most of previous work, our results hold on a more general edge arrival model. More specifically, we present (almost) optimal approximation algorithms for maximum coverage and minimum set cover problems in the streaming model with an (almost) optimal space complexity of $ ilde{O}(n)$, i.e., the space is {\em independent of the size of the sets or the size of the ground set of elements}. These results not only improve over the best known algorithms for the set arrival model, but also are the first such algorithms for the more powerful {\em edge arrival} model. In order to achieve the above results, we introduce a new general sketching technique for coverage functions: This sketching scheme can be applied to convert an $α$-approximation algorithm for a coverage problem to a $(1-\eps)α$-approximation algorithm for the same problem in streaming, or RAM models. We show the significance of our sketching technique by ruling out the possibility of solving coverage problems via accessing (as a black box) a $(1 \pm \eps)$-approximate oracle (e.g., a sketch function) that estimates the coverage function on any subfamily of the sets.

Motivation & Objective

• Address the limitations of prior streaming algorithms that rely on oracle access to entire sets, which ignores the cost of reading or storing large sets.
• Develop algorithms that work in the more general edge-arrival model, where set-element memberships arrive one by one, rather than sets arriving as units.
• Achieve almost optimal approximation ratios and space complexity (Õ(n)) for both maximum coverage and minimum set cover problems.
• Demonstrate that standard sketching approaches and access to noisy oracle estimators for coverage functions are insufficient for achieving good approximations.
• Establish theoretical limits by showing that the proposed space complexity is almost tight, even under favorable assumptions.

Proposed method

• Introduce a new general sketching technique that transforms any α-approximation offline algorithm into a (1−ε)α-approximation streaming algorithm.
• Use the sketching technique to maintain compact summaries of set families in a streaming fashion, enabling space-efficient computation without storing full sets.
• Apply the sketching method in a recursive, multi-pass streaming framework: iteratively remove covered elements and apply the sketching algorithm on the residual graph.
• Combine the sketching-based selection with a final greedy pass on the remaining uncovered elements to ensure constant-factor approximation.
• Use a hierarchical decomposition of the problem via parameterized thresholds (λ) to progressively reduce the number of uncovered elements across passes.
• Prove that the total space used is Õ(n/λ³), and with r passes, the total space becomes Õ(nm^{O(1/r)}), which is sublinear in m and nearly optimal.

Experimental results

Research questions

• RQ1 Can we design streaming algorithms for coverage problems that achieve near-optimal approximation ratios without assuming oracle access to entire sets?
• RQ2 Is it possible to achieve Õ(n) space complexity in the edge-arrival model, independent of the size of the sets or the ground set?
• RQ3 Can a general sketching technique be constructed to convert any offline α-approximation algorithm into a (1−ε)α-approximation streaming algorithm?
• RQ4 Why do standard sketching approaches fail for coverage problems, and what structural properties of coverage functions make them challenging to sketch?
• RQ5 Is access to a (1±ε)-approximate oracle for the coverage function sufficient to achieve good streaming approximations?

Key findings

• The proposed algorithm achieves a (1+ε)log m-approximation for the set cover problem with probability 1−1/n, using Õ(nm^{O(1/r)}) space.
• The space complexity is Õ(n), independent of the size of the sets or the ground set, making it nearly optimal and scalable to large inputs.
• The algorithm works in the more realistic edge-arrival model, where set-element memberships arrive in arbitrary order, unlike prior work restricted to set-arrival models.
• The sketching technique enables a (1−ε)α-approximation for any α-approximation offline algorithm, providing a general framework for streaming coverage problems.
• The paper proves that access to a (1±ε)-approximate oracle for the coverage function is insufficient to achieve good streaming approximations, highlighting the limitations of standard sketching.
• The space complexity is almost tight: no algorithm can achieve better than Õ(n) space for constant-factor approximation in the edge-arrival model, under standard assumptions.