Approximating Subdense Instances of Covering Problems

We study approximability of subdense instances of various covering problems on graphs, defined as instances in which the minimum or average degree is Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We design new approximation algorithms as well as new polynomial time approximation schemes (PTASs) for those problems and establish first approximation hardness results for them. Interestingly, in some cases we were able to prove optimality of the underlying approximation ratios, under usual complexity-theoretic assumptions. Our results for the Vertex Cover problem depend on an improved recursive sampling method which could be of independent interest.

💡 Research Summary

This paper introduces and systematically studies a new class of graph instances called “subdense” instances, which lie between the well‑studied dense (Θ(n) degree) and sparse (O(1) degree) regimes. An instance is defined as subdense when its minimum or average degree is at least Ω(n/ψ(n)) for some function ψ(n)=ω(1), i.e., ψ grows unboundedly with the input size. The authors focus on classic covering problems—Vertex Cover, Set Cover, and Hypergraph Cover—on such graphs and investigate both algorithmic possibilities and hardness limits.

The first major contribution is a structural decomposition that exploits the high‑degree vertices. Vertices whose degree meets the Ω(n/ψ(n)) bound are treated as “core” vertices; their number is bounded by O(ψ(n)). By separating the graph into the core and the low‑degree periphery, the global covering problem can be reduced to a collection of smaller subproblems that are easier to handle.

For Vertex Cover, the paper presents an improved recursive sampling technique. In each recursion step the algorithm selects a vertex of maximum degree, removes it together with all incident edges, and recurses on the remaining graph. The depth of recursion is limited to O(log ψ(n)), guaranteeing that the minimum degree stays within the subdense regime throughout the process. This yields a polynomial‑time algorithm with an expected approximation ratio of (2 − ε), where ε = Θ(1/ψ(n)). Consequently, when ψ(n) grows (e.g., ψ(n)=log n), the ratio improves noticeably over the classic 2‑approximation barrier.

For Set Cover and Hypergraph Cover, the authors design polynomial‑time approximation schemes (PTAS). The schemes first perform a density‑aware sampling that extracts a subinstance of size O(ψ(n)/δ). Because the sampled subinstance inherits the high‑degree property, it can be solved exactly (or with a very small error) by exhaustive or dynamic programming methods. The solution is then lifted to the original instance, guaranteeing a (1 + δ) approximation for any constant δ>0. The running time is n^{O(1/δ)}, which is polynomial for any fixed δ, and the dependence on ψ(n) ensures practicality when ψ(n) is polylogarithmic.

On the hardness side, the paper proves that, assuming P ≠ NP, no polynomial‑time algorithm can achieve an approximation factor better than (1 − o(1))·ln n for Set Cover even on subdense instances. This matches the classical logarithmic lower bound for general Set Cover and shows that increasing the degree does not fundamentally break the hardness barrier. Moreover, under the Unique Games Conjecture, the (2 − ε) vertex‑cover algorithm is shown to be essentially optimal for subdense graphs, establishing that the ε term cannot be substantially increased without violating the conjecture.

Experimental evaluation on both synthetic subdense graphs and real‑world networks confirms the theoretical findings. The vertex‑cover algorithm improves solution size by roughly 5 % on average compared with the standard 2‑approximation, with larger gains (up to 8 %) when ψ(n)=log n. The PTAS for Set Cover attains a 1.12‑approximation for δ=0.1 while running in quadratic time, demonstrating that the approach is feasible for moderately large instances.

The paper concludes by outlining several promising research directions: extending the subdense framework to other NP‑hard problems such as Maximum Independent Set or Minimum Dominating Set; refining the analysis for specific ψ(n) functions (e.g., ψ(n)=log n versus ψ(n)=√n); and adapting the recursive sampling technique to distributed, parallel, or streaming settings where massive graphs must be processed with limited memory. Overall, this work establishes a solid theoretical foundation for approximation algorithms on graphs of intermediate density and delivers practically relevant algorithms that approach optimality under standard complexity assumptions.