[Paper Review] Set Cover in Sub-linear Time

Piotr Indyk, Sepideh Mahabadi+3 more|arXiv (Cornell University)|Feb 10, 2019

Complexity and Algorithms in Graphs3 citations

TL;DR

This paper studies sub-linear time algorithms for the Set Cover problem, proposing a novel query-based approach that achieves nearly optimal approximation guarantees. It establishes tight query complexity bounds—showing that $ ilde{O}(m(n/k)^{1/(α-1)} + nk)$ queries suffice for an $α$-approximate cover, while proving a nearly-matching lower bound of $ ilde{Ω}(m(n/k)^{1/(2α)})$ for estimating the optimal cover size, even in the sub-linear regime.

ABSTRACT

We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of $m$ sets over $n$ elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in Har-Peled et al. [2016] to the sub-linear query model, that returns an $α$-approximate cover using $ ilde{O}(m(n/k)^{1/(α-1)} + nk)$ queries to the input, where $k$ denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of $k$, the required number of queries is $ ildeΩ(m(n/k)^{1/(2α)})$, even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require $Ω(nk)$ queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter $k$. On the other hand, we show that this bound is not optimal for larger values of the parameter $k$, as there exists a $(1+\varepsilon)$-approximation algorithm with $ ilde{O}(mn/k\varepsilon^2)$ queries. We show that this bound is essentially tight for sufficiently small constant $\varepsilon$, by establishing a lower bound of $ ildeΩ(mn/k)$ query complexity.

Motivation & Objective

• To investigate the feasibility and complexity of solving the Set Cover problem in sub-linear time using query access to the input.
• To establish nearly tight upper and lower bounds on the number of queries required for approximating the minimum set cover.
• To analyze the query complexity of both approximating the cover and verifying coverage, especially for small and large values of the optimal cover size $k$.
• To develop a probabilistic construction technique for hard instances that enables strong lower bounds for sub-linear algorithms.

Proposed method

• Adapting a streaming algorithm from [17] to the sub-linear query model, using an oracle-based access model with EltOf and SetOf oracles to retrieve set-element memberships.
• Designing a probabilistic construction of set systems where a small number of carefully chosen swaps can drastically reduce the minimum set cover size from $\alpha k$ to $k$, making such changes hard to detect without many queries.
• Applying Yao’s principle to prove lower bounds by constructing two distributions of instances (Yes and No) that are statistically hard to distinguish with few queries.
• Using a layered analysis based on urn models to bound the probability of detecting swaps in slabs, where each slab represents a potential modification to the set system.
• Establishing a lower bound of $\tilde{\Omega}(mn/k)$ for $(1+\varepsilon)$-approximation when $\varepsilon$ is a sufficiently small constant, showing the bound is nearly tight.
• Proving that verifying whether a given collection covers all elements requires $\Omega(nk)$ queries, highlighting the inherent difficulty of even basic coverage checks in sub-linear time.

Experimental results

Research questions

• RQ1 What is the minimum number of queries required to compute an $\alpha$-approximate set cover in sub-linear time?
• RQ2 Can we achieve better query complexity for large values of the optimal cover size $k$?
• RQ3 What is the query complexity of estimating the size of the minimum set cover, even without constructing the cover?
• RQ4 Is it possible to verify whether a given set collection covers the universe using sub-linear queries?
• RQ5 What probabilistic constructions can be used to prove strong lower bounds for sub-linear algorithms in combinatorial optimization?

Key findings

• An $\alpha$-approximate set cover can be computed using $\widetilde{O}(m(n/k)^{1/(α-1)} + nk)$ queries, matching the best known upper bound for this problem in the sub-linear model.
• A nearly-matching lower bound of $\widetilde{\Omega}(m(n/k)^{1/(2\alpha)})$ queries is proven for estimating the optimal cover size, even when only the size is required.
• The Cover Verification problem—checking whether a given set collection covers all elements—requires $\Omega(nk)$ queries, showing that even basic coverage checks are hard in sub-linear time.
• For $(1+\varepsilon)$-approximation with small constant $\varepsilon$, the algorithm requires $\widetilde{O}(mn/k\varepsilon^2)$ queries, and this bound is nearly tight due to a $\widetilde{\Omega}(mn/k)$ lower bound.
• The probabilistic construction of hard instances, where a small number of swaps can reduce the minimum cover size from $\alpha k$ to $k$, enables strong lower bounds by making modifications undetectable without many queries.
• The technique of using 'almost uniformly distributed' modifications in a structured set system provides a general framework that may be applicable to lower bounds in other combinatorial optimization problems.