Shortest two disjoint paths in conservative graphs

TL;DR

This work addresses finding two openly disjoint $s$-$t$ paths of minimum total weight in undirected graphs with conservative edge weights, a problem that becomes NP-hard when negative weights are allowed. By restricting negative edges to a constant number $c$ of trees, the authors develop a polynomial-time algorithm that combines flow techniques for separable solutions with a sophisticated recursion and dynamic-programming framework for non-separable cases. Key contributions include structural lemmas on how minimum-weight solutions interact with negative trees, an uncrossing lemma for merging partial path pairs, and a dynamic-programming method to compute partial solutions on the trees, all culminating in an overall running time of $O(n^{2c+9})$. This yields an XP algorithm parameterized by $c$ and opens questions about fixed-parameter tractability in $c$ and extensions to more terminals.

Abstract

We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph $G=(V,E)$ with edge weights $w:E \rightarrow \mathbb{R}$, two terminals $s$ and $t$ in $G$, find two internally vertex-disjoint paths between $s$ and $t$ with minimum total weight. As shown recently by Schlotter and Sebő (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, there are no cycles in $G$ with negative total weight. We propose a polynomial-time algorithm that solves the Shortest Two Disjoint Paths problem for conservative weights in the case when the negative-weight edges form a constant number of trees in $G$.

Figures (8)

• Figure 1: Illustration for Case C in the proof of Lemma \ref{['lem:uncrossing']}. Paths $P_1$ and $P_2$ are shown in blue, paths $Q_1$ and $Q_2$ in green, and edges of $T$ in purple. Closed walks $W$ (in Case C/1), $W_1$ and $W_2$ (in Case C/2) are depicted with coral red, dashed lines.
• Figure 2: Illustration for the proof of Claim \ref{['clm:Xmonotone']}. Edges within $T$ are depicted using solid lines, edges not in $T$ using dashed lines. Paths $Q_1$ and $Q_2$ are shown in blue and in green, respectively. We highlighted the closed walk $W$ in coral red.
• Figure 3: Illustration for the proof of Lemma \ref{['lem:starting']}. Edges within $T$ are depicted using solid lines, edges not in $T$ using dashed lines. Paths $Q_1$ and $Q_2$ are shown in blue and in green, respectively. We highlighted paths $T[a_1,z]$ and $T[a_2,y]$ in coral red.
• Figure 4: Illustration for the proof of Claim \ref{['clm:plain']}. Edges within $T$ are depicted using solid lines, edges not in $T$ using dashed lines. Paths $Q_1$ and $Q_2$ are shown in blue and in green, respectively. We highlighted the closed walk $W$ in coral red.
• Figure 5: Illustration for Case A in Lemma \ref{['lem:partsol-alg']}. Edges within $T$ are depicted using solid lines, edges not in $T$ using dashed lines. The figure assumes $Q_1=A_1 \cup T[x_1,u]$ (so $h=1$). Paths $Q_1$ and $Q_2$ are shown in blue and in green, respectively.

Theorems & Definitions (58)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Definition 4: Locally cheapest path pairs
• Corollary 6
• Lemma 7
• proof
• Lemma 8