Random Order Set Cover is as Easy as Offline

TL;DR

This paper studies OnlineSetCover under uniformly random element arrival (ROSetCover) and proposes LearnOrCover, a polynomial-time randomized algorithm that achieves an expected competitive ratio of $O(\log(mn))$, effectively matching the offline bound when $m$ is poly$(n)$. The core idea is to learn about the underlying set system as elements arrive, using a multiplicative-weights update to a fractional cover and online rounding to maintain feasibility, with a potential function that couples KL divergence to the optimal fractional solution with progress on the remaining uncovered weight. The authors also extend the framework to random-order covering integer programs (ROCIP), achieving the same $O(\log(mn))$ bound and preserving feasibility up to a constant factor via a simple post-processing step. Complementary lower bounds demonstrate that adversarial-order OnlineSetCover retains worst-case hardness $\Omega(\log m \log n)$ and stronger limitations persist in batched and extended settings, highlighting the distinct advantages of the random-order model. Overall, the work advances online learning-based approaches for RO problems, yielding near-optimal performance with modest memory and suggesting broad applicability to other RO sequential decision problems.

Abstract

We give a polynomial-time algorithm for OnlineSetCover with a competitive ratio of $O(\log mn)$ when the elements are revealed in random order, essentially matching the best possible offline bound of $O(\log n)$ and circumventing the $Ω(\log m \log n)$ lower bound known in adversarial order. We also extend the result to solving pure covering IPs when constraints arrive in random order. The algorithm is a multiplicative-weights-based round-and-solve approach we call LearnOrCover. We maintain a coarse fractional solution that is neither feasible nor monotone increasing, but can nevertheless be rounded online to achieve the claimed guarantee (in the random order model). This gives a new offline algorithm for SetCover that performs a single pass through the elements, which may be of independent interest.

Figures (1)

• Figure 1: Tight instance for buchbinder2009online in random order.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof : Proof of \ref{['thm:unit_cost_exp']}
• Claim 2.2
• proof
• Claim 2.3
• proof
• Theorem 3.1: Main Theorem
• Lemma 3.2: Potential Bounds