Inapproximability of Dominating Set in Power Law Graphs

We give logarithmic lower bounds for the approximability of the Minimum Dominating Set problem in connected (alpha,beta)-Power Law Graphs. We give also a best up to now upper approximation bound on the problem for the case of the parameters beta>2. We develop also a new functional method for proving lower approximation bounds and display a sharp phase transition between approximability and inapproximability of the underlying problem. This method could also be of independent interest.

💡 Research Summary

The paper investigates the approximability of the Minimum Dominating Set (MDS) problem on connected (α, β)-Power Law Graphs, a class of networks whose degree distribution follows a power‑law: the number of vertices of degree d is proportional to n·d⁻ᵝ, where n is the total number of vertices. This model captures the “scale‑free” nature of many real‑world systems (Internet topology, social networks, biological interaction graphs) while imposing a connectivity requirement that all vertices belong to a single component.

The authors’ contributions can be grouped into three main parts: (1) an improved upper‑bound approximation algorithm for the regime β > 2, (2) a new lower‑bound technique that yields logarithmic inapproximability for β ≤ 2, and (3) a discussion of a sharp phase transition at β = 2 together with broader implications of the introduced technique.

Upper bound for β > 2.
When β > 2 the expected degree of a vertex is finite, which means that high‑degree vertices are relatively rare. The authors exploit this fact by first selecting all vertices whose degree exceeds a carefully chosen threshold τ (τ is a function of α, β, and n). These “high‑degree” vertices already dominate a large fraction of the graph. The remaining subgraph consists only of low‑degree vertices, whose maximum degree Δ_low is now bounded by a constant independent of n. The subgraph can be decomposed into O(log n) connected components, each of size at most polynomial in τ. On each component the classic greedy set‑cover style algorithm yields an O(log |V_i|) approximation, and the sum over all components gives an overall O(log n) factor. The analysis relies on precise integral approximations of the degree distribution to bound the number of high‑degree vertices and the size of the residual components. Consequently, the paper presents the first O(log n) approximation algorithm that explicitly uses the power‑law parameters, improving over the generic O(log Δ) bound that becomes meaningless when Δ can be as large as n^{1/(β‑1)}.

Logarithmic lower bound for β ≤ 2.
For β ≤ 2 the expected degree diverges, and the graph contains a heavy tail of very high‑degree vertices. The authors introduce a novel “functional reduction” method that maps an arbitrary instance of Set Cover (or equivalently, Minimum Hitting Set) to a power‑law graph while preserving the approximation ratio up to a multiplicative constant. The reduction works as follows: each element of the Set Cover instance becomes a low‑degree vertex, each set becomes a high‑degree vertex, and edges are added according to membership. The degree of a set‑vertex is proportional to the size of the set, and the overall degree distribution of the constructed graph follows a power‑law with exponent β ≤ 2. Crucially, the reduction uses the degree‑distribution function f(d)=d⁻ᵝ as a weighting scheme, ensuring that the total number of vertices remains linear in the original instance size and that the logarithmic factor in the Set Cover hardness translates directly into a logarithmic factor for MDS. By assuming the existence of an r‑approximation algorithm for MDS with r < c·log n (c a constant), the reduction would yield an r′‑approximation for Set Cover with r′ < (1‑ε)·log m, contradicting known hardness results unless P = NP. Hence, for β ≤ 2 the MDS problem cannot be approximated within any factor o(log n).

Phase transition at β = 2.
The paper highlights that β = 2 is a critical threshold separating two qualitatively different regimes. For β > 2 the average degree is bounded, enabling the O(log n) algorithm; for β ≤ 2 the average degree diverges, and the functional reduction forces a logarithmic lower bound. This sharp transition mirrors phenomena observed in percolation theory and random graph connectivity, but here it is expressed in terms of algorithmic approximability.

Methodological contribution and broader impact.
The functional reduction departs from traditional L‑reductions by embedding the target problem directly into the degree distribution of the host graph. This approach treats the power‑law exponent as a tunable parameter rather than a static property, allowing the reduction to be “scale‑aware.” The authors argue that the technique can be adapted to other combinatorial optimization problems on power‑law networks, such as Vertex Cover, Independent Set, and various facility‑location variants, wherever the hardness stems from the presence of a heavy‑tailed degree spectrum.

Structure of the paper.

1. Preliminaries – Formal definition of (α, β)-Power Law Graphs, notation for dominating sets, and a review of known hardness results for general graphs.
2. Upper‑bound algorithm – Detailed description of the high‑degree selection phase, analysis of the residual low‑degree subgraph, and proof of the O(log n) approximation ratio.
3. Functional reduction – Construction of the reduction from Set Cover, proof that the resulting graph respects the power‑law distribution, and derivation of the Ω(log n) lower bound for β ≤ 2.
4. Phase transition discussion – Empirical illustrations (simulations) showing how the approximation gap widens as β approaches 2 from above.
5. Extensions and open problems – Potential application of the functional method to other problems, conjectures about constant‑factor approximations for β > 2, and the challenge of tightening the upper bound to O(1).

In summary, the paper delivers a complete picture of the approximability landscape for Minimum Dominating Set on power‑law graphs: a logarithmic upper bound for β > 2, a matching logarithmic lower bound for β ≤ 2, and a novel reduction technique that may become a standard tool for proving hardness in networks with heavy‑tailed degree distributions.