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| Disjoint paths problem | |
|---|---|
| Name | Disjoint paths problem |
| Field | Graph theory; Computer Science |
| Introduced | 1960s |
| Notable | Robert E. Tarjan; Neil Robertson; Paul D. Seymour; Noga Alon |
Disjoint paths problem.
The Disjoint paths problem asks whether, given a graph and a set of terminal pairs, there exist pairwise disjoint paths connecting each pair. Originating in combinatorial Graph theory and Algorithmic game theory contexts, the problem connects to major results by researchers associated with Princeton University, Bell Labs, Massachusetts Institute of Technology, and researchers such as Robert E. Tarjan and the team behind the Graph Minors Project at Princeton University.
Formally, given an undirected or directed graph G and a list of pairs (s1, t1), (s2, t2), ..., (sk, tk), the decision problem asks whether there exist k pairwise disjoint paths Pi each connecting si to ti. Classic formulations consider edge-disjointness or vertex-disjointness in graphs studied by groups at Stanford University, IBM, and universities involved in the EATCS community. The problem statement ties to foundational theorems like Menger's theorem and to structural results from the Graph Minors Project culminating in results by Neil Robertson and Paul D. Seymour.
Vertex-disjoint and edge-disjoint versions distinguish whether vertices (except terminals) or edges cannot be shared among paths; these variants were treated in seminal work by researchers affiliated with AT&T Bell Laboratories and the Institut des Hautes Études Scientifiques. Node-capacitated variants assign capacities to vertices, linking to flow formulations studied at Stanford University and in doctoral work at University of California, Berkeley. The k-disjoint paths problem fixes k as part of the input or as a parameter; fixed-k results relate to the Graph Minors Project and to parameterized complexity research by scholars at Carnegie Mellon University and Tel Aviv University such as Noga Alon.
The general problem is NP-complete in many formulations, a fact established in computational complexity literature associated with conferences like STOC and FOCS and labs at Bell Labs. For fixed k, the undirected k-disjoint paths problem is solvable in polynomial time via deep structural theory from the Graph Minors Project led by Neil Robertson and Paul D. Seymour. The directed version remains NP-complete even for small k, with hardness results published in venues involving researchers from Princeton University, Massachusetts Institute of Technology, and University of California, San Diego. Hardness reductions often reference classic NP-complete problems studied at Cornell University and MIT. Approximation hardness underpins connections to PCP theorems developed by researchers at Rutgers University and University of California, Berkeley.
Exact algorithms for special cases derive from constructive proofs in the Graph Minors Project by Neil Robertson and Paul D. Seymour, while combinatorial algorithms using max-flow and matching techniques trace to work by László Lovász and Robert E. Tarjan. Approximation algorithms for edge-disjoint routing in capacitated networks have been advanced by teams at IBM Research and Microsoft Research, building on randomized rounding methods associated with scholars at Princeton University and Stanford University such as Noga Alon. Fixed-parameter tractable (FPT) approaches parameterized by k leverage methods from parameterized complexity pioneered at Carnegie Mellon University and formalized in workshops at ICALP and ESA; notable contributors include researchers connected to Tel Aviv University and University of Edinburgh.
Planar graphs admit stronger results: the undirected disjoint paths problem in planar graphs has polynomial-time algorithms influenced by work by researchers at University of Waterloo and University of British Columbia, and exploits planar duality used in studies at Princeton University. Bounded treewidth and bounded genus graph classes yield FPT or polynomial-time solutions, with techniques developed in the Graph Minors Project and pursued at ETH Zurich and École Polytechnique Fédérale de Lausanne. Grid graphs and series-parallel graphs are classical special cases examined in algorithmic graph theory seminars at MIT and Stanford University.
Applications arise in optical network routing researched by teams at Bell Labs and AT&T Laboratories, in VLSI routing studied at Caltech and Carnegie Mellon University, and in traffic and evacuation planning explored by groups at Imperial College London and University College London. Practical implementations use integer programming and flow-based heuristics developed at Google and Huawei research labs, and leverage software libraries originating from projects at INRIA and University of California, Berkeley. Empirical evaluation often appears in proceedings of SIGCOMM, SODA, and INFOCOM where collaborators from Microsoft Research, IBM Research, and industrial partners present benchmarks and datasets.