Optimal Bridge, Twin Bridges and Beyond: Inserting Edges into a Road Network to Minimize the Constrained Diameters

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.