Given a road network modelled as a planar straight-line graph $G=(V,E)$ with $|V|=n$, let $(u,v)\in V\times V$, the shortest path (distance) between $u,v$ is denoted as $\delta_G(u,v)$. Let $\delta(G)=\max_{(u,v)}\delta_G(u,v)$, for $(u,v)\in V\times V$, which is called the diameter of $G$. Given a disconnected road network modelled as two disjoint trees $T_1$ and $T_2$, this paper first aims at inserting one and two edges (bridges) between them to minimize the (constrained) diameter $\delta(T_1\cup T_2\cup I_j)$ going through the inserted edges, where $I_j, j=1,2$, is the set of inserted edges with $|I_1|=1$ and $|I_2|=2$. The corresponding problems are called the {\em optimal bridge} and {\em twin bridges} problems. Since when more than one edge are inserted between two trees the resulting graph is becoming more complex, for the general network $G$ we consider the problem of inserting a minimum of $k$ edges such that the shortest distances between a set of $m$ pairs $P=\{(u_i,v_i)\mid u_i,v_i\in V, i\in [m]\}$, $\delta_G(u_i,v_i)$'s, are all decreased. The main results of this paper are summarized as follows: (1) We show that the optimal bridge problem can be solved in $O(n^2)$ time and that a variation of it has a near-quadratic lower bound unless SETH fails. The proof also implies that the famous 3-SUM problem does have a near-quadratic lower bound for large integers, e.g., each of the $n$ input integers has $\Omega(\log n)$ decimal digits. We then give a simple factor-2 $O(n\log n)$ time approximation algorithm for the optimal bridge problem. (2) We present an $O(n^4)$ time algorithm to solve the twin bridges problem, exploiting some new property not in the optimal bridge problem. (3) For the general problem of inserting $k$ edges to reduce the (graph) distances between $m$ given pairs, we show that the problem is NP-complete.

翻译：给定一个道路网络，建模为平面直线图 $G=(V,E)$，其中 $|V|=n$，令 $(u,v)\in V\times V$，$u$ 与 $v$ 之间的最短路径（距离）记为 $\delta_G(u,v)$。定义 $\delta(G)=\max_{(u,v)}\delta_G(u,v)$，其中 $(u,v)\in V\times V$，称为 $G$ 的直径。给定一个由两棵不相交的树 $T_1$ 和 $T_2$ 建模的非连通道路网络，本文首先旨在在两者之间插入一条或两条边（桥梁），以最小化通过插入边的（约束）直径 $\delta(T_1\cup T_2\cup I_j)$，其中 $I_j, j=1,2$ 是插入边的集合，且 $|I_1|=1$ 和 $|I_2|=2$。相应的问题分别称为“最优桥梁”问题和“双桥”问题。由于在两棵树之间插入多于一条边时，结果图变得更加复杂，对于一般网络 $G$，我们考虑插入最少 $k$ 条边的问题，使得一组 $m$ 个配对 $P=\{(u_i,v_i)\mid u_i,v_i\in V, i\in [m]\}$ 的最短距离 $\delta_G(u_i,v_i)$ 全部减小。本文的主要结果总结如下：(1) 我们证明最优桥梁问题可以在 $O(n^2)$ 时间内求解，并且其一个变体具有接近二次的下界，除非 SETH 失败。该证明还暗示著名的 3-SUM 问题对于大整数（例如，每个 $n$ 个输入整数具有 $\Omega(\log n)$ 十进制位数）具有接近二次的下界。随后，我们给出一个简单的因子-2 $O(n\log n)$ 时间近似算法求解最优桥梁问题。(2) 我们提出一个 $O(n^4)$ 时间算法求解双桥问题，利用了不同于最优桥梁问题中的某些新性质。(3) 对于插入 $k$ 条边以减小 $m$ 个给定配对之间的（图）距离的一般问题，我们证明该问题是 NP 完全的。