A weighted graph is called stable if the maximum weight of an integral matching equals the cost of a minimum-weight fractional vertex cover. We address the following question: how can we modify a given unstable graph in the least intrusive manner in order to achieve stability? Previous works have addressed stabilization through addition or deletion of the smallest possible number of edges/vertices. In this work we investigate the following more fine-grained additive stabilization strategy: given a graph $G=(V,E)$ with unit edge weights; find non-negative $c\in {\mathbb{R}}^E$ with minimum ${\sum }_e{c}_e$ such that adding $c_e$ to the unit edge weight of each $e\in E$ yields a stable graph.

We provide the first super-constant hardness of approximation results for any graph stabilization problem: (i) unless the current best-known algorithm for the densest- $k$-subgraph problem can be improved, there is no $o\left(\right|V{|}^{1∕24})$-approximation for additive stabilizers; (ii) the additive stabilizer problem has no $o(log|V\left|\right)$ approximation unless $P=NP$.

On the algorithmic side, we present (iii) a polynomial time algorithm with approximation factor at most $\sqrt{\left|V\right|}$ for a super-class of the instances generated in our hardness proofs, (iv) an algorithm to solve min additive stabilizer in factor-critical graphs exactly in polynomial time, and (v) an algorithm to solve min additive stabilizer in arbitrary graphs exactly in time exponential in the size of the Tutte set. Our main tools are the Gallai–Edmonds decomposition and structural results for the problem that reduce the continuous decision domain to a discrete decision domain.

如果整数匹配的最大权重等于最小权重的分数顶点覆盖的成本，则加权图称为稳定图。我们解决以下问题：如何以最小的介入方式修改给定的不稳定图以获得稳定性？以前的工作已经通过增加或删除尽可能少的边/顶点来解决稳定性问题。在这项工作中，我们研究了以下更细粒度的添加剂稳定策略：给定一个图$G=（V，Ë）$具有单位边缘权重；找出非负数$C\in {\mathbb{[R}}^{Ë}$ 最少 ${\sum }_{Ë}{C}_{Ë}$ 这样添加 $C_{Ë}$ 到每个的单位边缘重量 $Ë\in Ë$ 产生稳定的图形。

对于任何图形稳定问题，我们都提供近似结果的第一超常数硬度：（i）除非当前最著名的算法用于最稠密的- $ķ$-subgraph问题可以改善，没有$Ø（|V{|}^{\mathrm{1个}∕24}）$-添加剂稳定剂的近似值；（ii）添加剂稳定剂问题没有$Ø（日志\left|V\right|）$ 近似值除非 $P=ñP$。

在算法方面，我们提出（iii）最多具有近似因子的多项式时间算法 $\sqrt{\left|V\right|}$对于在我们的硬度证明中生成的超类实例，（iv）一种精确地在多项式时间内求解关键因子图中的最小加法稳定剂的算法，以及（v）一种精确地求解任意图中的最小加法稳定剂的算法。 Tutte集合大小的时间指数。我们的主要工具是Gallai–Edmonds分解和针对该问题的结构结果，这些问题可将连续决策域减少为离散决策域。