2-Disjoint Shortest Paths (2-DSP)

• 2-DSP is a special case of the k-Disjoint Shortest Paths problem, where each terminal pair must have a individually shortest and vertex-disjoint path.
• The methodology employs techniques like BFS layering, geometric embeddings, and dynamic programming to efficiently isolate and compute valid paths.
• Key results show that while 2-DSP is polynomial-time solvable in both directed and undirected graphs, extending these methods to k≥3 remains challenging.

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 (Akmal et al., 2022). 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 (Lochet, 2019). 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 (Pilipczuk et al., 6 May 2025).

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 (Akmal et al., 2022). In the unweighted undirected formulation, path length is the number of edges, and the condition becomes

$(s_1,t_1)$0

where $(s_1,t_1)$1 is graph distance (Lochet, 2019). For $(s_1,t_1)$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 (Lochet, 2019). 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 (Björklund et al., 2018).

Variant	Requirement	Representative source
Classical 2-DSP	Each $(s_1,t_1)$3 is a shortest path for its own pair; paths are node-disjoint	(Akmal et al., 2022, Lochet, 2019)
S2DP / SDPP	Two disjoint paths minimizing total length	(Björklund et al., 2018, Chakraborty et al., 20 Mar 2025)
2-RSP	Two edge-disjoint paths minimizing total cost under a delay bound	(Guo et al., 2015)
Activation 2-DP	Two internally node-disjoint $(s_1,t_1)$4-$(s_1,t_1)$5 paths under activation cost	(Nutov et al., 2023)

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 $(s_1,t_1)$6 admits a polynomial-time algorithm; later work extended polynomial-time solvability to any fixed $(s_1,t_1)$7 in undirected graphs (Lochet, 2019). In directed graphs, 2-DSP is also polynomial-time solvable, whereas for fixed $(s_1,t_1)$8 on general directed graphs the complexity remains open (Akmal et al., 2022).

The broader parameterized picture is less benign. On undirected graphs, $(s_1,t_1)$9-DSP is W[1]-hard with respect to $(s_2,t_2)$0, and, assuming ETH, there is no algorithm with running time $(s_2,t_2)$1 (Bentert et al., 2020). 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 $(s_2,t_2)$2, while 2-DSP on directed graphs is polynomial-time solvable (Akmal et al., 2022). 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 (Akmal et al., 2022).

3. Core algorithmic frameworks

One influential general framework reduces undirected $(s_2,t_2)$3-DSP to a colored shortest-path problem on a $(s_2,t_2)$4-shortest graph (Lochet, 2019). Starting from BFS layerings rooted at the sources $(s_2,t_2)$5, 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_2,t_2)$6 becomes a path of color $(s_2,t_2)$7. 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 (Lochet, 2019). This yields a polynomial-time algorithm for fixed $(s_2,t_2)$8.

For the unweighted undirected two-pair case, a different approach uses geometry rather than BFS coloring. Each vertex is embedded as

$(s_2,t_2)$9

and shortest paths are confined to $P_1,P_2$0-diamonds between the endpoint coordinates (Bentert et al., 2020). 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 $P_1,P_2$1 (Bentert et al., 2020). This geometric lens improves the running time for $P_1,P_2$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 (Gottschau et al., 2018). 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-$P_1,P_2$3 $P_1,P_2$4-DSP is polynomial-time solvable (Akmal et al., 2022). On planar graphs with positive edge weights, the parameterized complexity improves substantially: Planar Disjoint Shortest Paths admits an algorithm with running time

$P_1,P_2$5

and the paper explicitly notes that this parameter dependence is better than the state-of-the-art $P_1,P_2$6 for the planar disjoint paths problem without shortestness (Pilipczuk et al., 6 May 2025).

The planar result is structurally significant because general undirected $P_1,P_2$7-DSP is W[1]-hard, yet its planar restriction becomes fixed-parameter tractable (Pilipczuk et al., 6 May 2025). 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 (Pilipczuk et al., 6 May 2025). For $P_1,P_2$8, 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 $P_1,P_2$9-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 $P_i$0-path together with a shortest $P_i$1-path (Akmal et al., 2022). 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_i$2 connecting $P_i$3 and $P_i$4 while minimizing $P_i$5; this is the S2DP or SDPP viewpoint rather than strict 2-DSP (Björklund et al., 2018). 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 (Datta et al., 2021). These results are highly relevant historically, but they do not solve the strict per-pair-shortest formulation.

A second nearby line is $P_i$6-Disjoint Restricted Shortest Paths, where one minimizes total cost under a separate global delay bound and requires edge-disjointness (Guo et al., 2015). A third replaces additive path costs by node-based activation costs: Activation 2-DP asks for two internally node-disjoint $P_i$7-$P_i$8 paths minimizing

$P_i$9

which is again structurally close to 2-DSP but not the same optimization problem (Nutov et al., 2023).

There are also single-source multipath variants. Given a fixed source $k$0, one can ask for $k$1 edge-disjoint paths of minimum total cost from $k$2 to every other target $k$3, together with a sparse preserver of size $k$4 (Bilò et al., 2021). For $k$5, 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 (Akmal et al., 2022). Fixed-$k$6 undirected $k$7-DSP is polynomial-time solvable (Lochet, 2019), yet the parameterized problem is W[1]-hard in general (Bentert et al., 2020). On planar graphs with positive edge weights, the problem becomes fixed-parameter tractable with running time $k$8 (Pilipczuk et al., 6 May 2025). The main unresolved frontier singled out in the surveyed work is the complexity of $k$9-DSP for fixed $P_1,\dots,P_k$0 on general directed graphs (Akmal et al., 2022).

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.