A Polynomial-Time Algorithm for the Next-to-Shortest Path Problem on Positively Weighted Directed Graphs

TL;DR

This work resolves the long-standing open question of whether a next-to-shortest $s\to t$ path can be found in polynomial time for directed graphs with positive edge weights by introducing a reduction to $(s,t)$-layered graphs and proving a central Key Lemma. The approach decomposes any not-shortest $s\to t$ path into a middle subpath flanked by two forward-disjoint paths, enabling a construction that combines two forward paths with a shortest middle segment to yield a next-to-shortest path. The authors develop a layered-graph algorithm NextSP-Layered, with a complexity bound of $O(|V|^4|E|^3\log |V|)$, and provide a reduction pipeline through $(s,t)$-straight graphs to layered graphs, ensuring polynomial-time solvability for all positively weighted directed graphs. This advances the theory of near-shortest paths and provides a practical framework for computing strictly second-shortest paths in directed, positively weighted networks. The results have implications for network design and routing where near-optimal alternatives are required efficiently, and they settle a nearly 30-year-old open problem in the affirmative under $P\neq NP$-style assumptions.

Abstract

Given a graph and a pair of terminals $s$, $t$, the next-to-shortest path problem asks for an $s\!\to \!t$ (simple) path that is shortest among all not shortest $s\!\to \!t$ paths (if one exists). This problem was introduced in 1996, and soon after was shown to be NP-complete for directed graphs with non-negative edge weights, leaving open the case of positive edge weights. Subsequent work investigated this open question, and developed polynomial-time algorithms for the cases of undirected graphs and planar directed graphs. In this work, we resolve this nearly 30-year-old open problem by providing an algorithm for the next-to-shortest path problem on directed graphs with positive edge weights.

Figures (7)

• Figure 1: Illustration of the reduction: (a) a subgraph of $G$, and (b) the corresponding subgraph in $G'$ obtained by modification.
• Figure 2: An illustration of Back-Edge Decomposition.
• Figure 3: An illustration of $(X',X),(Y',Y)$ and $P^{*}$ in \ref{['lem:key']}
• Figure 4: An example of \ref{['def:updown']}. $P$ is a path from $u_1$ to $u_7$, and $V^{\operatorname{inter}}(P_1^{*},P_2^{*};P)=\lbrace u_1,u_2,\ldots,u_7\rbrace$. $P_{u_1\to u_2},P_{u_2\to u_3},P_{u_3\to u_4}$ and $P_{u_5\to u_6}$ are gap-paths. However, $P_{u_2\to u_3}$ and $P_{u_3\to u_4}$ contain no gap-vertices. $P_{u_4\to u_5}$ is a down-path. $P_{u_6\to u_7}$ is an up-path.
• Figure 5: An illustration of proof for \ref{['lem:up-vertex-only']}.

Theorems & Definitions (43)

• Theorem 1.1
• Definition 3.1: $s\!\!\to\!\! t$ Not-Shortest and Next-to-Shortest Paths
• Definition 3.2: The Next-to-Shortest Path Problem
• Lemma 3.1
• proof
• Lemma 3.2: tholey2012linear
• Definition 4.1: $(s,t)$-Layered Graph
• Theorem 4.1
• Definition 4.2: Shortcut Edge
• Definition 4.3: $\pi$