---
title: 2-Disjoint Shortest Paths (2-DSP)
url: https://www.emergentmind.com/topics/2-disjoint-shortest-paths-2-dsp
type: topic
---

# 2-Disjoint Shortest Paths (2-DSP)

to=arxiv_search.search ＿俺去也json രൂപത്തിലുള്ള  大发快三计划:
{"query":"\"Disjoint Shortest Paths\" 2-DSP arXiv", "max_results": 10, "sort_by": "relevance"}
to=arxiv_search.search  彩神争霸如何json ,大香蕉:
{"query":"\"2-Disjoint Shortest Paths\" arXiv", "max_results": 10, "sort_by": "relevance"}
to=arxiv_search.search  彩神争霸官网json:
{"query":"shortest two disjoint paths arXiv", "max_results": 10, "sort_by": "relevance"}
2-Disjoint Shortest Paths (2-DSP) is the two-pair special case of the \(k\)-Disjoint Shortest Paths problem: given a graph and two terminal pairs \((s_1,t_1)\) and \((s_2,t_2)\), the task is to find two vertex-disjoint paths \(P_1,P_2\) such that each \(P_i\) is a shortest path between its own endpoints [2211.07042]. In the strict formulation, shortestness is imposed separately on each pair, not on the sum of the two path lengths, and disjointness is usually vertex-disjointness, often phrased as internal vertex-disjointness when the terminals are distinct [1912.10486]. The subject lies at the intersection of shortest-path geometry, disjoint routing, structural graph theory, and parameterized complexity, and its modern development combines exact polynomial algorithms for the two-pair case with broader fixed-parameter results for planar and undirected graph classes [2505.03353].

## 1. Definition, scope, and terminology

In the weighted formulation used for congested shortest paths, graphs have positive edge weights, and a \(k\)-DSP instance asks for paths \(P_1,\dots,P_k\) such that each \(P_i\) is a shortest path from \(s_i\) to \(t_i\) and the paths are node-disjoint [2211.07042]. In the unweighted undirected formulation, path length is the number of edges, and the condition becomes
\[
|P_i| = d(s_i,t_i)\quad\text{for all } i\in[k],
\]
where \(d(x,y)\) is graph distance [1912.10486]. For \(k=2\), this yields the classical 2-DSP decision problem.

Two conventions are important. First, the literature discussed here is primarily about **vertex-disjoint** paths, not merely edge-disjoint paths [1912.10486]. Second, the phrase “shortest two disjoint paths” is not terminologically uniform across adjacent subliteratures: some papers use it for a min-sum objective, while others use it for the stricter per-pair shortestness condition [1806.07586].

| Variant | Requirement | Representative source |
|---|---|---|
| **Classical 2-DSP** | Each \(P_i\) is a shortest path for its own pair; paths are node-disjoint | [2211.07042], [1912.10486] |
| **S2DP / SDPP** | Two disjoint paths minimizing total length | [1806.07586], [2503.16336] |
| **2-RSP** | Two edge-disjoint paths minimizing total cost under a delay bound | [1504.05519] |
| **Activation 2-DP** | Two internally node-disjoint \(s\)-\(t\) paths under activation cost | [2307.12646] |

A recurring misconception is to identify all of these formulations with 2-DSP. They are closely related, but they optimize different objects and obey different feasibility constraints. The strict 2-DSP condition requires each path individually to remain shortest, which is stronger than minimizing the total length of a disjoint pair.

## 2. Complexity landscape

The two-pair case is tractable. In undirected graphs, 2-DSP was introduced by Eilam-Tzoreff, who proved that the case \(k=2\) admits a polynomial-time algorithm; later work extended polynomial-time solvability to any fixed \(k\) in undirected graphs [1912.10486]. In directed graphs, 2-DSP is also polynomial-time solvable, whereas for fixed \(k\ge 3\) on general directed graphs the complexity remains open [2211.07042].

The broader parameterized picture is less benign. On undirected graphs, \(k\)-DSP is W[1]-hard with respect to \(k\), and, assuming ETH, there is no algorithm with running time \(f(k)\cdot n^{o(k)}\) [2007.12502]. This establishes a sharp distinction between the tractability of the two-pair case and the difficulty of scaling the problem with the number of terminal pairs.

A further contrast concerns the relationship with the classical disjoint paths problem. In general directed graphs, the ordinary directed disjoint paths problem is NP-complete even for \(k=2\), while 2-DSP on directed graphs is polynomial-time solvable [2211.07042]. The literature explicitly emphasizes this contrast as evidence that the shortest-path requirement can make the problem easier in some settings, because it imposes strong metric structure on feasible solutions [2211.07042].

## 3. Core algorithmic frameworks

One influential general framework reduces undirected \(k\)-DSP to a colored shortest-path problem on a \(k\)-shortest graph [1912.10486]. Starting from BFS layerings rooted at the sources \(s_1,\dots,s_k\), the reduction deletes edges that do not connect consecutive levels for at least one BFS. Each surviving edge is given one or more colors, and a shortest path for pair \((s_i,t_i)\) becomes a path of color \(i\). The resulting algorithms exploit two structural notions: **bi-coloured components**, which localize where shortest paths of different colors can overlap, and **blind paths**, which control non-conflicting interactions between differently colored shortest paths [1912.10486]. This yields a polynomial-time algorithm for fixed \(k\).

For the unweighted undirected two-pair case, a different approach uses geometry rather than BFS coloring. Each vertex is embedded as
\[
\pos{v} = (\dist(s_1,v),\dist(s_2,v)) \in \mathbb{N}^2,
\]
and shortest paths are confined to \(\ell_\infty\)-diamonds between the endpoint coordinates [2007.12502]. The algorithm analyzes how the drawings of two shortest paths can intersect in this grid, reduces the relevant cases to a bounded number of DAG instances, and solves 2-DSP in time \(O(nm)\) [2007.12502]. This geometric lens improves the running time for \(k=2\) and gives a particularly transparent explanation of why the two-pair case is special.

For the undirected **edge-disjoint** shortest-path variant with non-negative lengths, including zero-length edges, another framework constructs a partially oriented expansion into a weakly acyclic mixed graph and then applies dynamic programming over its component order [1809.03820]. This result is not the vertex-disjoint 2-DSP problem itself, but it is part of the same algorithmic ecosystem: shortestness is encoded through orientation and layering, and disjointness is recovered by dynamic programming in a structured auxiliary graph.

## 4. Structured graph classes and parameterized results

Several graph classes admit stronger statements than the general undirected setting. For DAGs and undirected graphs, fixed-\(k\) \(k\)-DSP is polynomial-time solvable [2211.07042]. On planar graphs with positive edge weights, the parameterized complexity improves substantially: Planar Disjoint Shortest Paths admits an algorithm with running time
\[
2^{O(k\log k)}\cdot n^{O(1)},
\]
and the paper explicitly notes that this parameter dependence is better than the state-of-the-art \(2^{O(k^2)}\) for the planar disjoint paths problem without shortestness [2505.03353].

The planar result is structurally significant because general undirected \(k\)-DSP is W[1]-hard, yet its planar restriction becomes fixed-parameter tractable [2505.03353]. The underlying techniques use geodesic Steiner trees, ring decompositions, shortest-path DAGs, and homology feasibility to control how geodesics can wind around one another in a planar embedding [2505.03353]. For \(k=2\), this FPT algorithm specializes to a polynomial-time planar algorithm, although the dedicated two-pair algorithms are stronger in practice.

Related structural work studies shortest-path systems rather than disjointness directly. In DAGs, if every 3-set of vertices lies on a shortest path, then some shortest \(st\)-path contains all vertices; in undirected graphs the analogous statement holds with 4-sets, and in directed graphs the best available result is a roundtrip analogue using a shortest \(st\)-path together with a shortest \(ts\)-path [2211.07042]. These results are not algorithms for 2-DSP, but they show that shortest-path families can exhibit rigid global structure once local overlap conditions become strong.

## 5. Adjacent problems and common sources of confusion

A substantial neighboring literature concerns **shortest two disjoint paths** in the sense of minimizing the total length of a disjoint pair. In that formulation, one seeks two vertex-disjoint paths \(P_1,P_2\) connecting \((s_1,t_1)\) and \((s_2,t_2)\) while minimizing \(\ell(P_1)+\ell(P_2)\); this is the S2DP or SDPP viewpoint rather than strict 2-DSP [1806.07586]. Algebraic algorithms based on permanents modulo powers of two place this min-sum two-path problem in randomized parallel complexity classes and interpret it through disjoint cycles [2106.00714]. These results are highly relevant historically, but they do not solve the strict per-pair-shortest formulation.

A second nearby line is \(k\)-Disjoint Restricted Shortest Paths, where one minimizes total cost under a separate global delay bound and requires edge-disjointness [1504.05519]. A third replaces additive path costs by node-based activation costs: Activation 2-DP asks for two internally node-disjoint \(s\)-\(t\) paths minimizing
\[
\sum_{v\in V}\max_{e\in \delta_F(v)} c_e^v,
\]
which is again structurally close to 2-DSP but not the same optimization problem [2307.12646].

There are also single-source multipath variants. Given a fixed source \(s\), one can ask for \(p\) edge-disjoint paths of minimum total cost from \(s\) to every other target \(t\), together with a sparse preserver of size \(p(n-1)\) [2106.12293]. For \(p=2\), this recovers the Suurballe–Tarjan setting of two edge-disjoint minimum-total-cost paths from one source, which is different from the two-pair strict-shortest problem but frequently appears beside it in networking applications.

## 6. Present status and open directions

The current picture is sharply stratified. The strict two-pair problem is polynomial-time solvable in both undirected and directed graphs [2211.07042]. Fixed-\(k\) undirected \(k\)-DSP is polynomial-time solvable [1912.10486], yet the parameterized problem is W[1]-hard in general [2007.12502]. On planar graphs with positive edge weights, the problem becomes fixed-parameter tractable with running time \(2^{O(k\log k)}\cdot n^{O(1)}\) [2505.03353]. The main unresolved frontier singled out in the surveyed work is the complexity of \(k\)-DSP for fixed \(k\ge 3\) on general directed graphs [2211.07042].

A second frontier concerns variant proliferation. The literature surrounding 2-DSP now includes strict per-pair shortestness, min-sum shortest disjoint paths, restricted and bicriteria formulations, activation-cost models, and single-source multipath preservers. This suggests that “2-DSP” is best treated as the core member of a wider family of metric-disjoint routing problems rather than as an isolated decision problem.

A plausible implication is that the modern theory of 2-DSP is no longer centered on the existence of a polynomial algorithm for two pairs; that milestone has long been passed. The active questions concern how shortestness interacts with planarity, directionality, homology, geometric embeddings, and parameterization. In that sense, 2-DSP functions both as a solved base case and as the prototype from which the contemporary theory of disjoint shortest-path systems continues to develop.

Source: https://www.emergentmind.com/topics/2-disjoint-shortest-paths-2-dsp