关于添加链路以最小化有效图电阻问题的非子模性
On the non-submodularity of the problem of adding links to minimize the effective graph resistance
AI总结：
针对加链最小化有效图电阻的 NP-hard 问题，本文通过图族反例证明其子模率可趋零，并以10节点图揭示贪心算法无质量保证且与最优解比率低至0.878。
摘要：
我们考虑如下优化问题：向给定网络添加 $k$ 条链路，使所得有效图电阻尽可能小。该问题最近被证明是 NP-hard，因此通过暴力方法获得最优解需要指数级数量的计算步骤，对于任何具有现实规模的图都不可行。所以，在这种情况下通常使用简单的贪心算法来获得最优解的近似。已知如果所考虑的问题是子模的，贪心解的质量就可以得到保证。然而，我们面对的优化问题已知并不是子模的。对于这种情况，可以使用由子模率 $γ$ 刻画的广义子模性概念。在具有广义子模性时，也存在一个作为 $γ$ 的函数的性能界。本文给出一个图族示例，其中子模率趋近于零，这意味着贪心算法的解质量无法得到保证。此外，我们表明贪心算法并不总能产生最优解，并证明即使对于一个具有 10 个节点的小图，最优解与贪心解之间的比率也可能低至 0.878。
英文摘要：
We consider the optimisation problem of adding $k$ links to a given network, such that the resulting effective graph resistance is as small as possible. The problem was recently proven to be NP-hard, such that optimal solutions obtained with brute-force methods require exponentially many computation steps and thus are infeasible for any graph of realistic size. Therefore, it is common in such cases to use a simple greedy algorithm to obtain an approximation of the optimal solution. It is known that if the considered problem is submodular, the quality of the greedy solution can be guaranteed. However, it is known that the optimisation problem we are facing, is not submodular. For such cases one can use the notion of generalized submodularity, which is captured by the submodularity ratio $γ$. A performance bound, which is a function of $γ$, also exists in case of generalized submodularity. In this paper we give an example of a family of graphs where the submodularity ratio approaches zero, implying that the solution quality of the greedy algorithm cannot be guaranteed. Furthermore, we show that the greedy algorithm does not always yield the optimal solution and demonstrate that even for a small graph with 10 nodes, the ratio between the optimal and the greedy solution can be as small as 0.878.
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