最密子图问题的线性时间近似方案
A Linear-Time Approximation Scheme for the Densest Subgraph Problem
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中文总结 AI 辅助
本文针对最密子图问题，提出首个真正线性时间的(1-ε)近似算法，还为最密至少k子图问题给出近线性时间的(1/2 - ε)近似算法，性能接近理论难度下界。
中文摘要 AI 辅助
在无向最密子图问题（Densest Subgraph Problem, DSG）中，目标是输出给定图G的顶点子集S，使得子图诱导边集E(S)与S的顶点数之比|E(S)|/|S|最大化。该问题在理论与实践中均得到广泛研究，存在自然的高效精确算法，以及近线性时间的(1-ε)近似算法。然而，所有已知的近似方案均会引入图规模或其他参数的对数因子，这引发了一个问题：是否能对所有ε>0得到线性时间的(1-ε)近似？本文通过提供一个时间复杂度为O((n+m)/ε³ log(1/ε))的(1-ε)近似算法给出了肯定答案，其中m和n分别为G的边数和顶点数。据我们所知，这是该问题首个真正的线性时间近似方案（当ε>0为常数时）。我们的算法结合了基于流公式的分配与结构性切割引理，该引理允许我们逐步切割图的“稀疏”部分，同时近乎保留最密子图，从而将繁重计算转移到更小的实例中，最终得到上述运行时间。我们的框架还为最密至少k子图问题（Densest At-Least-k Subgraph Problem）提供了(1/2 - ε)近似，该问题除最大化密度外，还要求子图至少有k个顶点，对应算法运行时间为O(((n+m) log²n log(1/ε))/ε)，这几乎匹配已知的1/2近似难度，且在近线性时间内运行。
英文摘要
In the undirected \emph{Densest Subgraph Problem (DSG)} the goal is to output a subset $S$ of vertices of a given graph $G$ that maximizes the quantity $|E(S)|/|S|$, where $E(S)$ is the set of edges in the subgraph induced by $S$. The problem is well studied in both theory and practice, and it admits natural efficient exact algorithms, as well as near-linear time algorithms with a $(1-\varepsilon)$ approximation ratio. However, all previously-known approximation schemes incur logarithmic factors in the size of the graph or other parameters of the graph. This raises the question of whether a linear time $(1-\varepsilon)$-approximation can be obtained for all $\varepsilon>0$. We answer this question affirmatively by providing a $(1-\varepsilon)$-approximation algorithm running in time $O\left(\frac{n+m}{\varepsilon^3}\log \frac{1}{\varepsilon}\right)$, where $m$ and $n$ are respectively the number of edges and vertices of $G$. To the best of our knowledge, this is the first truly linear-time approximation scheme for the problem (when $\varepsilon>0$ is a constant). Our algorithm uses assignments arising from a flow-based formulation together with a structural carving lemma. This lemma allows us to progressively carve "sparse" parts of the graph while nearly preserving the densest subgraph, allowing us to shift heavy computations to smaller instances, which eventually yields the mentioned runtime. Our framework also yields a $(1/2 -\varepsilon)$-approximation for the \emph{Densest At-Least-$k$ Subgraph Problem}, where in addition to maximizing the density, we require the subgraph to have at least $k$ vertices. Our algorithm runs in time $O\left( \frac{(n+m) \log^2 n \log \frac{1}{\varepsilon}}{\varepsilon} \right)$. This nearly matches the known $1/2$ approximation hardness while running in near-linear time.
