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arXiv 1512.08070cs.DM
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最小费用2-边连通生成子图问题的6/5界

Toward a 6/5 Bound for the Minimum Cost 2-Edge Connected Spanning Subgraph Problem

Sylvia Boyd, Philippe Legault

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摘要：

给定一个具有非负边代价$c\in {\mathbb R}^{E}$的完全图$K_{n}=(V, E)$，$2EC$问题是寻找$K_{n}$的最小代价的2-边连通生成多子图的问题. $2EC$的线性规划松弛$\text{2EC}^{\text{LP}}$的完整性间隙$α\text{2EC}$已被证明是$\frac{6}{5}$，尽管目前我们只知道$\frac{6}{5}\leqα\text{2EC}\leq\frac{3}{2}$。本文利用$\text{2EC}^{\text{LP}}$的解的结构和凸组合的概念，得到了$α\text{2EC}$的改进的上界.我们把我们的努力集中在家庭$J$的半整数解决方案，似乎给最大的完整性差距$\text{2EC}^{\text{LP}}$。我们成功地证明了猜想$α\text{2EC} = \frac{6}{5}$对任何由某些$x^{*}\in J$优化的代价函数都是成立的。

英文摘要：

Given a complete graph $K_{n}=(V, E)$ with non-negative edge costs $c\in {\mathbb R}^{E}$, the problem $2EC$ is that of finding a 2-edge connected spanning multi-subgraph of $K_{n}$ of minimum cost. The integrality gap $α\text{2EC}$ of the linear programming relaxation $\text{2EC}^{\text{LP}}$ for $2EC$ has been conjectured to be $\frac{6}{5}$, although currently we only know that $\frac{6}{5}\leqα\text{2EC}\leq\frac{3}{2}$. In this paper, we explore the idea of using the structure of solutions for $\text{2EC}^{\text{LP}}$ and the concept of convex combination to obtain improved bounds for $α\text{2EC}$. We focus our efforts on a family $J$ of half-integer solutions that appear to give the largest integrality gap for $\text{2EC}^{\text{LP}}$. We successfully show that the conjecture $α\text{2EC} = \frac{6}{5}$ is true for any cost functions optimized by some $x^{*}\in J$.

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