匹配增强问题的 $59/33$ 割-LP 保证
A $59/33$ Cut-LP Guarantee for Matching Augmentation
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- Institute of Computer Science, University of Augsburg（奥格斯堡大学计算机科学研究所）
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中文总结 AI 辅助
本文通过新的结构分析，证明匹配增强问题的LP引导DFS算法达到59/33的积分间隙，无需新算法步骤，并推广到森林增强问题。
中文摘要 AI 辅助
匹配增强问题（MAP）要求找到一个最小基数的单位成本边集，该边集与一个零成本匹配一起构成一个2-边连通的多重生成图。我们研究标准的割松弛。Bamas、Drygala和Svensson提出了一种特别简单的LP引导算法：计算一个极端最优解，运行一个优先考虑大LP坐标的深度优先搜索，并最优地增强所得的DFS树。我们对Bamas–Drygala–Svensson LP引导的DFS算法给出了新的结构分析。该分析将残差上行链路的精确原始-对偶恒等式与一个秩界相结合，该秩界度量相对于单位值骨架的分数支撑。结果是，对于每个根和每个与LP优先级一致的确定性平局打破顺序，算法返回的解的成本至多为 $\frac{59}{33}c(x^*) - \frac{25}{33} = \left(2 - \frac{7}{33}\right)c(x^*) - \frac{25}{33} \approx 1.788c(x^*) - 0.758$，其中 $x^*$ 是割LP的最优解。因此，该松弛的积分间隙至多为 $59/33 \approx 1.788$。不需要新的算法步骤；改进是分析性的。残差上行链路问题的精确打包证书产生一个带有打包松弛项的成本恒等式，而秩定理则界定相对于单位值骨架的分数支撑。区域分类解释了非树边，而双割恒等式处理自洞。作为直接推论，相同的 $59/33 \approx 1.788$ 界在最小值情形下也适用于森林增强问题。证明是自包含的，除了一个关于最小割向量维数的定理。
英文摘要
The Matching Augmentation Problem (MAP) asks for a minimum-cardinality set of unit-cost edges that, together with a zero-cost matching, forms a 2-edge-connected spanning multigraph. We study the standard cut relaxation. Bamas, Drygala, and Svensson proposed a particularly simple LP-guided algorithm: compute an extreme optimum, run a depth-first search that prioritizes large LP coordinates, and augment the resulting DFS tree optimally. We give a new structural analysis of the Bamas--Drygala--Svensson LP-guided DFS algorithm. The analysis combines an exact primal--dual identity for the residual uplink problem with a rank bound that measures fractional support relative to the unit-valued skeleton. The result is that for every root and every deterministic tie-breaking order consistent with the LP priorities, the algorithm returns a solution of cost at most $\frac{59}{33}c(x^*)-\frac{25}{33}=\left(2-\frac7{33}\right)c(x^*)-\frac{25}{33}\approx1.788c(x^*)-0.758$, where $x^*$ is an optimum of the cut LP. Consequently, the integrality gap of the relaxation is at most $59/33\approx1.788$. No new algorithmic step is required; the improvement is analytical. The exact packing certificate for the residual uplink problem yields a cost identity with a packing-slack term, while a rank theorem bounds fractional support relative to the unit-valued skeleton. A regional classification accounts for the non-tree edges, and a two-cut identity handles self-holes. As a direct corollary, the same $59/33\approx1.788$ bound holds for Forest Augmentation in the minimum-value regime. The proof is self-contained apart from one theorem on the dimension of minimum-cut vectors.
