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

TL;DR

The paper studies how inserting a small number of edges into a road network (modeled as a weighted planar graph) can minimize diameters and pairwise distances. It provides a precise $O(n^2)$ algorithm for the optimal bridge problem (connecting two trees by a single edge), a conditional near-quadratic lower bound for the one-bridge variant under SETH, and a $O(n\log n)$-time 2-approximation; it extends to a twin-bridges setting with an $O(n^4)$ algorithm. It also establishes strong complexity boundaries by proving NP-completeness for the general Reducing Distances Between Pairs problem and relates these results to known geometric bridge problems via reductions to COV and 3-SUM. Overall, the work delineates the tractability frontier for strategically augmenting road networks to reduce diameters and targeted distances, with implications for network design and geometric graph theory.

Abstract

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 $δ_G(u,v)$. Let $δ(G)=\max_{(u,v)}δ_G(u,v)$, 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 $δ(T_1\cup T_2\cup I_j)$ 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 $P=\{(u_i,v_i)\mid u_i,v_i\in V, i\in [m]\}$, $δ_G(u_i,v_i)$'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 $Ω(\log n)$ 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.

Figures (5)

• Figure 1: An example for the reduction from COV to the one-bridge decision problem. The vertical paths, which should all be on the $Y$-axis, are drawn for a better visualization.
• Figure 2: An example where two paths $(a,b,c)$ and $(d,e,f)$ need to be connected into a spanning tree. $be$ and $cf$ are positioned by rotating a segment of length $n$ at $b$ and $e$ slightly to achieve that $|cf|=1-\epsilon$. Adding $(b,e)$ would result in an optimal diameter of $2n+1$; with the greedy method of adding the shortest edge between the two paths, $(c,f)$ is added to achieve a diameter of $4n+1-\varepsilon$. (The distances in the figure are not measured geometrically, due to the space constraint.)
• Figure 3: An illustration for the twin bridges problem.
• Figure 4: An example for the twin bridges problem in which the two optimal bridges must intersect. The two input trees are paths: $T_1=(x,a,b,y)$ and $T_2=(z,c,d,w)$.
• Figure 5: An illustration for the reduction from Vertex Cover on Planar 2-Connected Cubic Graphs to RDBP.

Theorems & Definitions (11)

• Theorem 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.2
• Theorem 3.3
• Corollary 3.1
• Theorem 3.4
• Lemma 4.1
• Theorem 4.1