Approximate k-Cover in Hypergraphs: Efficient Algorithms, and Applications

Given a weighted hypergraph $\mathcal{H}(V, \mathcal{E} \subseteq 2^V, w)$, the approximate $k$-cover problem seeks for a size-$k$ subset of $V$ that has the maximum weighted coverage by \emph{sampling only a few hyperedges} in $\mathcal{E}$. The problem has emerged from several network analysis applications including viral marketing, centrality maximization, and landmark selection. Despite many efforts, even the best approaches require $O(k n \log n)$ space complexities, thus, cannot scale to, nowadays, humongous networks without sacrificing formal guarantees. In this paper, we propose BCA, a family of algorithms for approximate $k$-cover that can find $(1-\frac{1}{e} -\epsilon)$-approximation solutions within an \emph{$O(\epsilon^{-2}n \log n)$ space}. That is a factor $k$ reduction on space comparing to the state-of-the-art approaches with the same guarantee. We further make BCA more efficient and robust on real-world instances by introducing a novel adaptive sampling scheme, termed DTA.

💡 Research Summary

The paper tackles the weighted hypergraph k‑cover problem, which asks for a set of k vertices that maximizes the total weight of hyperedges incident to the chosen vertices. This formulation underlies many large‑scale network‑analysis tasks such as influence maximization, landmark selection, and k‑dominating set. Traditional solutions rely on the classic greedy algorithm for submodular maximization, guaranteeing a (1 − 1/e)‑approximation. However, to apply the greedy algorithm one must keep all sampled hyperedges in memory, leading to a space requirement of O(k n log n) – prohibitively large when the number of hyperedges grows quadratically or exponentially with n.

The authors introduce a new framework called BCA (Bounded Coverage Algorithms). BCA uses a random‑oracle that can generate hyperedges according to their weight distribution, but it never stores the full “sketch” of all generated hyperedges. Instead, it maintains a reduced sketch E_r, a compact subset of the samples, together with a dynamically updated upper‑bound function f(S, d_S, E_r) that estimates the maximum possible coverage any k‑set can achieve given the current sketch. A threshold z is set on this bound; once the bound falls below z, the algorithm stops sampling and discards non‑essential hyperedges from E_r. This mechanism guarantees that the number of stored hyperedges never exceeds O(ε⁻² n log n), independent of k, while still providing a (1 − 1/e − ε)‑approximation with probability at least 1 − δ.

On top of BCA the authors build an adaptive sampling scheme named DTA (Dynamic Threshold Adaptation). DTA monitors the empirical coverage of the current sketch and the gap between lower‑ and upper‑bound estimates. When the estimates converge, DTA reduces the sampling rate, thereby preventing unnecessary growth of the sketch. Conversely, during early iterations it samples aggressively to explore the solution space. The only user‑controlled parameter is ε; all other thresholds are derived automatically, making DTA robust across different graph structures and weight models (e.g., uniform, trivalency, perturbed WC).

Theoretical contributions include:

1. Proof that BCA returns a set \hat S with Cov_w(\hat S) ≥ (1 − 1/e − ε)·OPT with probability 1 − δ.
2. Space complexity O(ε⁻² n log n) and time complexity O(ε⁻² k n), both independent of the total number of hyperedges.
3. Extension of the framework to the budgeted version of k‑cover with minimal modifications.

Empirical evaluation covers three representative applications: influence maximization (IM), landmark selection (LMS), and k‑dominating set. The authors compare DTA‑BCA against state‑of‑the‑art algorithms such as IMM, DSSA, SSA, PreX Hedge, and Y‑alg. Results show that DTA‑BCA reduces the sketch size by up to 1000× and runs up to 10× faster while achieving comparable or better influence spread, landmark coverage, and domination ratios. Notably, DTA‑BCA successfully processes billion‑node graphs on a single machine, where existing methods either run out of memory or require distributed infrastructures.

In summary, the paper delivers a practically viable solution to large‑scale approximate k‑cover: a space‑optimal BCA framework that eliminates the dependence on k in memory usage, and an adaptive DTA sampler that automatically balances accuracy and resource consumption. The combination of rigorous approximation guarantees, optimal space bounds, and strong experimental performance positions BCA/DTA as a compelling alternative for any application that can be modeled as weighted hypergraph coverage.