Vertex Disjoint Paths with Congestion

• Vertex Disjoint Paths with Congestion is a relaxation of the strict vertex-disjoint paths problem, allowing vertices to be shared by up to c paths.
• The research elaborates on formal models like (k,c)-Congestion Routing and DEDP, distinguishing local versus global congestion constraints and their algorithmic benefits.
• Complexity results reveal tractable regimes in acyclic graphs with slack parameters and NP-hard cases under strict constraints, highlighting the nuanced challenges in congestion routing.

Searching arXiv for papers on directed/vertex-disjoint paths with congestion and related relaxations. Searching for the batch vertex-disjoint paths paper to distinguish strict disjointness from congestion-relaxed models. Vertex Disjoint Paths with Congestion is a relaxation of the classical vertex-disjoint paths problem in which the routed paths need not be pairwise disjoint, but every vertex is allowed to lie on at most a prescribed number $c$ of paths. In the standard directed formulation, the input consists of a digraph $D$, an integer $c$, and terminal pairs $(s_1,t_1),\dots,(s_k,t_k)$; the task is to find paths $P_1,\dots,P_k$ such that each $P_i$ connects $s_i$ to $t_i$ and each vertex of $D$ appears in at most $c$ many paths. The case $D$0 is the ordinary directed vertex-disjoint paths problem, while larger $D$1 interpolate between strict disjointness and unconstrained overlap (Amiri et al., 2016, Kawarabayashi et al., 14 Jul 2025, Bentert et al., 16 Jul 2025).

1. Formal problem models

A standard formalization is $D$2-Congestion Routing: given a digraph $D$3 and $D$4 demand pairs $D$5, decide whether there exists a $D$6-routing, that is, a family $D$7 where each $D$8 links $D$9 to $c$0 and no vertex lies on more than $c$1 of the paths. In this setting, the congestion of a vertex is the number of routed paths containing it. The same model appears under closely related names such as Directed Disjoint Paths with Congestion, $c$2-Disjoint Directed Paths with Congestion $c$3, and $c$4 in different parts of the literature (Amiri et al., 2016, Kawarabayashi et al., 14 Jul 2025, Bentert et al., 16 Jul 2025).

The central parameter is the allowed overlap. When $c$5, the problem is the classical vertex-disjoint paths problem. When $c$6, the congestion constraint becomes vacuous, because a vertex may be used by all $c$7 paths. Several papers therefore analyze the high-congestion regime through the slack parameter

$c$8

which measures distance from the trivial all-overlapping case (Amiri et al., 2016, Amiri et al., 2020, Akmal et al., 2022).

A distinct but related relaxation is Disjoint Enough Directed Paths (DEDP). Here the input is a digraph $c$9, requests $(s_1,t_1),\dots,(s_k,t_k)$0, and integers $(s_1,t_1),\dots,(s_k,t_k)$1 and $(s_1,t_1),\dots,(s_k,t_k)$2, and the goal is to find paths satisfying the requests such that at most $(s_1,t_1),\dots,(s_k,t_k)$3 vertices occur in at least $(s_1,t_1),\dots,(s_k,t_k)$4 paths, while all other vertices occur in at most $(s_1,t_1),\dots,(s_k,t_k)$5 paths. Equivalently, with $(s_1,t_1),\dots,(s_k,t_k)$6, the requirement is that there exists a set $(s_1,t_1),\dots,(s_k,t_k)$7 of size at least $(s_1,t_1),\dots,(s_k,t_k)$8 such that every vertex in $(s_1,t_1),\dots,(s_k,t_k)$9 is used by at most $P_1,\dots,P_k$0 paths. This is a global congestion relaxation rather than a local bound at every vertex (Lopes et al., 2019).

Model	Constraint	Specialization
$P_1,\dots,P_k$1-Congestion Routing	every vertex is on at most $P_1,\dots,P_k$2 paths	$P_1,\dots,P_k$3 gives vertex-disjoint paths
DEDP	at least $P_1,\dots,P_k$4 vertices are on at most $P_1,\dots,P_k$5 paths	$P_1,\dots,P_k$6 gives local congestion bound $P_1,\dots,P_k$7 everywhere
batch-$P_1,\dots,P_k$8DP	paths are strictly vertex-disjoint for each query	not a congestion model

The batch-$P_1,\dots,P_k$9DP line is often confused with congestion relaxations because it studies sharing, but the sharing there is algorithmic rather than combinatorial: the paths remain strictly vertex-disjoint, and the paper explicitly states that it does not study congestion in the usual sense of allowing multiple paths to share vertices or edges with bounded load (Yuan et al., 23 Feb 2025).

2. Relations to classical disjointness, congestion, and Steiner-type routing

The classical directed disjoint paths problem asks for pairwise vertex-disjoint paths connecting prescribed terminal pairs. In congestion language, it is exactly the case $P_i$0. This identification is used repeatedly in the literature: the special case $P_i$1 of $P_i$2 is the usual Directed $P_i$3-Linkage / Vertex Disjoint Paths problem, and the special case $P_i$4 of $P_i$5-SPC or $P_i$6-Disjoint Shortest Paths with Congestion-$P_i$7 is the corresponding vertex-disjoint shortest-path problem (Kawarabayashi et al., 14 Jul 2025, Akmal et al., 2022, Amiri et al., 2020).

DEDP was introduced specifically to interpolate between strict disjointness and local congestion. If $P_i$8, then vertices in the good region are used by at most one path, so the paths are pairwise vertex-disjoint there; if moreover $P_i$9, then every vertex must be used by at most one path, and DEDP becomes exactly Directed Disjoint Paths. At the other extreme, if $s_i$0, equivalently $s_i$1, then every vertex must obey the local congestion bound $s_i$2, so DEDP becomes Directed Disjoint Paths with Congestion $s_i$3. If $s_i$4, the good region must be avoided entirely, and the paper states that this case corresponds to a parameterized form of Steiner Network (Lopes et al., 2019).

Shortest-path variants preserve the same boundary cases. In $s_i$5-SPC, when $s_i$6 the problem is easy because each shortest path can be computed independently, while when $s_i$7 the problem is exactly $s_i$8-Disjoint Shortest Paths. The same interpolation appears in the DAG-specific formulation of $s_i$9-Disjoint Shortest Paths with Congestion-$t_i$0, which treats $t_i$1 as the operative parameter in the high-congestion regime (Akmal et al., 2022, Amiri et al., 2020).

A recurring conceptual distinction is between local and global overlap control. The standard congestion model constrains every vertex individually. DEDP instead allows arbitrary congestion on an unspecified part of the graph, provided that a large subset of size at least $t_i$2 remains $t_i$3-bounded. This suggests two different senses in which paths may be “disjoint enough”: either uniformly everywhere, or only on a large well-behaved region (Lopes et al., 2019).

3. Complexity landscape on acyclic and general digraphs

For directed acyclic graphs, the high-congestion regime admits a precise positive result. For every fixed $t_i$4, $t_i$5-Congestion Routing on acyclic digraphs can be solved in time $t_i$6. The key structural lemma shows that when $t_i$7, a $t_i$8-routing exists if and only if every demand pair is individually reachable and some proper subset of the demands of size $t_i$9 has a $D$0-routing. This yields a reduction to a subinstance of at most $D$1 terminal pairs and leads to the $D$2-time algorithm (Amiri et al., 2016).

The same paper establishes strong negative evidence that this dependence on $D$3 is close to optimal. For any fixed integer $D$4, $D$5-Congestion Routing is W[1]-hard when parameterized by $D$6, and, assuming ETH, cannot be solved in time $D$7. Rephrased in the high-congestion parametrization, $D$8-Congestion Routing is W[1]-hard parameterized by $D$9, and assuming ETH cannot be solved in time $c$0 (Amiri et al., 2016).

Later work strengthened the hardness picture by showing that the vertex-congestion problem remains W[1]-hard for every congestion $c$1 even under severe structural restrictions: the input digraph can be acyclic, contain no $c$2 as a butterfly minor, contain no acyclic $c$3-grid as a butterfly minor, and have ear anonymity at most $c$4. An analogous theorem holds for the edge-congestion variant on acyclic digraphs of maximum undirected degree $c$5 excluding an acyclic $c$6-wall as a weak immersion (Kawarabayashi et al., 14 Jul 2025).

At the classical NP-completeness scale, a sharp threshold is known for fixed congestion. The directed disjoint paths problem with congestion $c$7 is NP-complete for any constant $c$8 and any $c$9. This refutes a conjecture that the congestion-two case might be polynomial-time solvable for any constant number of terminal pairs. The same work isolates the first unresolved frontier around half-integral routing: the case $D$00 is polynomial-time solvable, $D$01 is NP-complete by the general theorem, and $D$02 remains open (Bentert et al., 16 Jul 2025).

DEDP exhibits a different complexity profile because its congestion notion is global. For any fixed $D$03, if $D$04 and $D$05, then DEDP is NP-complete for every fixed $D$06 and $D$07, and $D$08-DEDP is W[1]-hard in DAGs for every fixed $D$09. Moreover, DEDP is W[1]-hard with parameter $D$10 for every fixed $D$11, even on acyclic digraphs where all source vertices are the same (Lopes et al., 2019).

4. Tractable regimes and structural methods

Beyond the acyclic high-congestion algorithm, several tractable islands are known under structural graph restrictions. DEDP is solvable in time

$D$12

on digraphs of directed tree-width at most $D$13. The proof extends the Johnson–Robertson–Seymour–Thomas framework through $D$14-guarded sets, arboreal decompositions, itineraries, and a lemma stating that every solution is $D$15-limited, meaning that inside any $D$16-guarded set the solution has at most $D$17 weak components (Lopes et al., 2019).

For the dual parameters of DEDP, an XP algorithm runs in time

$D$18

obtained by guessing a set $D$19 of size $D$20, checking whether $D$21 is $D$22-viable, and reducing larger $D$23 to the $D$24 case by duplicating vertices in $D$25. More strongly, $D$26-DEDP admits a kernelization algorithm running in time

$D$27

that either outputs a solution or an equivalent instance with at most

$D$28

vertices; hence $D$29-DEDP is FPT. The bypassing lemma for congested vertices is the central technical step (Lopes et al., 2019).

Semicomplete digraphs support a different FPT regime for the standard local congestion model. On semicomplete digraphs, $D$30-DDP is fixed-parameter tractable parameterized by $D$31 restricted to instances satisfying $D$32. The proof adapts the directed-pathwidth-plus-irrelevant-vertex methodology from edge-disjoint paths, replacing free-arc arguments by a congestion-based pigeonhole principle: when $D$33, two vertices each used by exactly $D$34 paths must lie on a common routed path. The method is essentially tight for this framework, since the same paper gives counterexamples with no irrelevant vertex when $D$35 (Gomes et al., 28 Apr 2025).

The polynomial-time solvability of the half-integral case $D$36 also depends on structural decomposition. The algorithm uses bounded directed tree-width dynamic programming when possible, and otherwise applies the directed grid theorem to extract a large cylindrical wall, invokes a routing theorem that either gives a half-integral linkage or a small separator, and then repeatedly uncrosses and contracts 2-separations until the remaining graph has bounded directed tree-width. The difficult bookkeeping is handled through feasible routings at contracted regions and special subroutines for $D$37-DP with $D$38 and $D$39 (Bentert et al., 16 Jul 2025).

5. Shortest-path congestion variants

A substantial subliterature studies shortest-path analogues of vertex-disjoint paths with congestion. In $D$40-Disjoint Shortest Paths with Congestion-$D$41, the input graph has positive edge weights and the task is to find paths $D$42 such that each $D$43 is a shortest path from $D$44 to $D$45 and every vertex is used by at most $D$46 paths. On DAGs, this problem can be solved in time $D$47, where $D$48. The reduction lemma states that when $D$49, the instance is solvable if and only if every terminal pair is connected and some subinstance on $D$50 terminal pairs is solvable for congestion $D$51 (Amiri et al., 2020).

That algorithm is complemented by strong hardness. For every constant $D$52, the congested shortest-path problem is W[1]-hard with respect to the parameter $D$53, and for any fixed $D$54 it cannot be solved in time $D$55 unless ETH fails. The paper also shows that $D$56-EDSP does not admit any $D$57 algorithm even on planar DAGs (Amiri et al., 2020).

A different route to polynomial-time solvability when $D$58 is constant comes from local-to-global theorems for shortest paths. For DAGs, if $D$59-DSP can be solved in time $D$60, then $D$61-SPC can be solved in time

$D$62

For undirected graphs, the corresponding bound is

$D$63

yielding polynomial-time solvability for constant $D$64 once combined with an algorithm for undirected $D$65-DSP (Akmal et al., 2022).

The structural mechanism is expressed through max-congestion nodes. If $D$66, then each max-congestion node is avoided by exactly $D$67 solution paths. Observation $D$68 states that if $D$69, then any set of $D$70 max-congestion nodes is contained in some solution path. In the undirected setting this leads to a heavy-path lemma: if $D$71, then any solvable instance has a solution in which some solution path contains all max-congestion nodes. In the directed setting the best available conclusion is weaker: if $D$72, then some two solution paths contain all max-congestion nodes, and this does not currently yield an analogous reduction to $D$73-DSP (Akmal et al., 2022).

6. Adjacent directions, structural theory, and common misconceptions

The strict vertex-disjoint paths literature remains central because many congestion results are phrased as relaxations of it. Examples include FPT algorithms for planar directed $D$74-Vertex-Disjoint Paths with running time $D$75, and the planar disjoint paths algorithm with running time $D$76 based on treewidth reduction, irrelevant vertices, and Schrijver-style algebraic or homological methods. These works do not introduce congestion parameters, but they supply decomposition tools—alternating cycles, ring components, treewidth reduction, unique linkage phenomena—that reappear in more permissive routing models (Cygan et al., 2013, Lokshtanov et al., 2020).

Wall-based min-max theory is likewise adjacent rather than identical. Recent results on connecting vertex sets to walls prove alternatives between a small vertex set hitting all relevant paths and a large subwall supporting many pairwise disjoint paths with endpoints on distinct nails. The overlap control is structural—clean systems, pure linkages, in-series linkages—rather than a per-vertex congestion bound (Bruhn et al., 22 Jun 2026).

Approximation with bounded overlap has also been studied in optimization versions such as MaxEDP. On graphs with feedback vertex set number $D$77, one can route $D$78 pairs with congestion

$D$79

improving the classical Raghavan–Thompson dependence on $D$80. The same paper gives an exact algorithm for MaxNDP running in time $D$81, while proving that MaxNDP is W[1]-hard parameterized by $D$82 (Fleszar et al., 2016).

A frequent misconception is that any work on “sharing” in disjoint-path routing is about congestion. The batch-$D$83DP framework of ShareDP is an explicit counterexample: it studies strict vertex-disjoint paths for many source–target pairs, using shared split-graph representations, tagged bidirectional BFS, and shared updates of $D$84, $D$85, and $D$86. The paper states that the shared resources are algorithmic computations and storage, not path overlap, and that the paths themselves remain strictly disjoint except at the two terminals of each query (Yuan et al., 23 Feb 2025).

Taken together, these results show that Vertex Disjoint Paths with Congestion is not a single problem family but a spectrum of routing models. The standard local model constrains every vertex by a uniform capacity $D$87; DEDP relaxes this to a large well-behaved region; shortest-path variants add geodesic constraints; semicomplete, planar, bounded-width, and acyclic classes each admit their own techniques. The unifying theme is that relaxing disjointness by bounded overlap can create algorithmic leverage, but usually only in regimes controlled by slack parameters such as $D$88, by structural graph classes, or by global restrictions on where congestion is allowed (Amiri et al., 2016, Lopes et al., 2019, Gomes et al., 28 Apr 2025).