covering number analysis

This work proposes a novel local computation algorithm (LCA) for the set cover problem that significantly reduces query complexity while preserving approximation quality. The key innovation lies in an aggressive input sparsification strategy combined with a “backtrack-and-update” mechanism, which enables the algorithm to dynamically revise decisions made in earlier recursive calls. This enhances solution concentration and curtails redundant computation. Theoretical analysis demonstrates that the proposed method lowers the query complexity from Δ^{O(log Δ)} to f^{O(log Δ)}; moreover, when f = polylog Δ, the complexity is further improved to Δ^{O(log log Δ)}, substantially outperforming existing approaches.

This paper studies the minimum-cost set cover problem under a fixed constant $ r $ of monotone submodular constraints: given a ground set $ N $, a cost function $ c: N o mathbb{R}_+ $, $ r $ monotone submodular functions $ f_i $, and thresholds $ b_i $, find a minimum-cost subset $ S subseteq N $ satisfying $ f_i(S) geq b_i $ for all $ i $. To overcome the bottleneck where classical algorithms’ approximation ratios degrade with $ r $, we propose the first bi-criteria randomized approximation algorithm. Our method integrates LP relaxation, weighted covering function techniques, and structural properties of deletion-closed systems. In expectation, the solution cost is at most $ alpha cdot mathrm{OPT} $, while achieving coverage ratio $ 1 - 1/e^alpha - varepsilon $. For weighted covering functions, we obtain an approximation ratio of $ (1+varepsilon)frac{e}{e-1}(1+eta) $, breaking the $ r $-dependent logarithmic lower bound.

This paper addresses the problem of designing vector-valued set-to-vector mapping functions that *exactly* preserve set inclusion: $S subseteq T iff F(S) leq F(T)$ (coordinate-wise). To this end, it introduces the notion of *Monotone and Separating* (MAS) functions, establishes necessary and sufficient conditions for their existence, and derives tight lower bounds on the minimal embedding dimension. For infinite ground sets, it constructs novel models satisfying a relaxed MAS property together with Hölder continuity. Methodologically, the approach unifies set function theory with neural network design, yielding a universal approximator architecture that inherently enforces monotonicity. Experiments demonstrate that the proposed model significantly outperforms standard set encoders—lacking explicit inclusion priors—across multiple set containment prediction tasks, thereby validating the effectiveness and generalization advantage of the introduced inductive bias.

This paper investigates the computational complexity of locally bijective covering problems for graphs with semi-edges, focusing on deciding whether an input graph covers a given target multigraph—allowing loops, multiple edges, and semi-edges—with one or two vertices. We provide the first systematic complexity classification of this generalized covering problem: (i) we establish that semi-edges introduce intrinsic hardness, rendering edge-mapping non-negligible; (ii) we fully classify the P vs. NP-complete dichotomy for all single- and two-vertex target graphs; (iii) we prove that covering is NP-complete for almost all two-vertex target graphs with semi-edges—even when restricted to regular or bipartite targets; and (iv) we precisely characterize all polynomial-time solvable cases. Our approach integrates combinatorial graph theory, modeling via local bijective homomorphisms, and carefully constructed NP-hardness reductions tailored to semi-edge structures.

This work addresses parameterized approximation algorithms for Vertex Cover and 3-Hitting Set. Methodologically, it introduces a novel randomized branching paradigm grounded in an equivalence between the algorithm’s recursive structure and a binary stochastic process. Leveraging a type-theoretic adaptation of Sanov’s theorem, the framework performs large-deviation analysis on bivariate recurrence relations, yielding an analytically tractable master theorem for asymptotic running time. Contribution-wise, this is the first unified theoretical framework providing rigorous approximation-ratio–dependent guarantees across multiple approximation factors. It substantially improves worst-case time complexity over prior deterministic branching approaches and overcomes fundamental analytical limitations inherent in traditional branching analysis. The framework establishes a general methodology for characterizing the asymptotic performance of parameterized approximation algorithms, bridging stochastic analysis and combinatorial optimization.

This work addresses the dynamic set cover problem, which requires efficiently maintaining an approximately optimal solution under frequent insertions and deletions of elements—a setting where existing theoretical algorithms lack empirical validation. For the first time, we present a systematic experimental evaluation of state-of-the-art dynamic algorithms based on greedy strategies, originating from GKKP (2017), SU (2023), and SUZ (2024). By simplifying complex subroutines and introducing a tunable parameter β to balance solution quality against computational efficiency, we enable practical deployment of these methods. Extensive experiments on real-world datasets reveal significant differences among the algorithms in terms of cover size, update time, and amortized overhead, offering the first empirical guidance for selecting and configuring dynamic set cover algorithms in practical applications.

This work addresses the exact algorithmic complexity of the $k$-Set Cover problem, where the universe contains $n$ elements and each set has size at most $k$. By leveraging refined combinatorial analysis and advanced techniques in exponential-time algorithm design, the authors develop a new exact algorithm that significantly improves the running time for all sufficiently large $k$. Specifically, they reduce the time complexity from $2^{(1 - 0.929/k)n}$ to $2^{(1 - 1/k + O(1/k^{3/2}))n}$, thereby enhancing the constant factor in the exponent. This result establishes the current best-known exact algorithm for large $k$, surpassing the previous bound by Björklund (STACS 2010).

This study addresses the geometric maximum coverage problem: given a set of geometric objects and a set of points (or a region), select at most $k$ objects to maximize the number of covered points (or total covered volume). By integrating shallow-cell complexity analysis, VC-dimension theory, parameterized algorithms, and geometric decomposition techniques, the work achieves the first improvements over the classical $1 - 1/e$ approximation ratio for various geometric objects—including pseudo-disks, fat rectangles, and same-size fat triangles—under the assumption of linear 2-shallow cell complexity or constant VC-dimension. It also develops more efficient parameterized approximation schemes for small $k$ and establishes an EPTAS for the continuous variant. Furthermore, the paper proves APX-hardness and the infeasibility of PTAS under certain settings.

This work investigates parameterized approximation bounds for Partial Set Cover and Maximum Coverage in set systems with bounded VC dimension. By introducing structural parameters such as the shatter function exponent and downward intersection complexity, it establishes the first hardness result showing that Partial Set Cover admits no $(2-\delta)$-approximation in FPT time when VC dimension is at least 7, unless $\text{FPT} = \text{W[1]}$. Conversely, under bounded shatter function exponent, a $(k+1)$-approximation guarantee is recovered and extended to weighted settings, multi-criteria objectives, and matroid constraints. Leveraging parameterized reductions and structural characterizations, the study designs a $2^{O(\Gamma k \log k)} N$-time algorithm for Weighted Partial Set Cover and an EPAS running in $2^{\tilde{O}(kd/\varepsilon)} N^{O(1)}$ time for Weighted CC-MaxSAT under bounded VC dimension.

This work proposes a novel architecture based on adaptive feature fusion and dynamic reasoning to address the limited generalization of existing methods in complex scenarios. By incorporating a multi-scale context-aware module and a learnable strategy for selecting inference paths, the approach significantly enhances model robustness and accuracy on out-of-distribution data. Extensive experiments demonstrate that the proposed method consistently outperforms state-of-the-art models across multiple benchmark datasets, exhibiting particularly strong adaptability under low-resource and high-noise conditions. The study not only offers a new perspective for open-world learning but also establishes a technical foundation for designing efficient and scalable intelligent systems.