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| Pathwidth | |
|---|---|
| Name | Pathwidth |
| Field | Graph theory |
| Introduced | 1980s |
| Notable | Neil Robertson, Paul D. Seymour, Robin Thomas |
Pathwidth Pathwidth is a graph invariant measuring how closely a graph resembles a path, capturing linear arrangement constraints and playing a central role in structural graph theory, algorithm design, and combinatorics. It refines notions of tree-likeness used by researchers at institutions such as Massachusetts Institute of Technology, University of Waterloo, Princeton University, and University of Oxford, and it interacts with major results from figures like Neil Robertson, Paul D. Seymour, and Robin Thomas. Pathwidth has influenced algorithmic work at organizations including AT&T Bell Laboratories, Microsoft Research, and Google.
Pathwidth is defined via a path decomposition: an ordered sequence of vertex sets (bags) whose union covers the graph and such that each edge has both endpoints in some bag, with the bags containing any given vertex forming a contiguous block along the sequence. The width equals the size of the largest bag minus one; the pathwidth of a graph is the minimum width over all path decompositions. Foundational expositions appear in texts by authors affiliated with Princeton University Press, Cambridge University Press, and lecture notes from courses at Stanford University, Harvard University, and University of Cambridge. Early studies connected pathwidth to combinatorial results by Paul Erdős, Endre Szemerédi, and applied work by Richard Karp.
Pathwidth is bounded below by treewidth and is at least as large as the treewidth of the same graph; conversely, any graph of treewidth k has pathwidth at most k times functions studied in structural theorems by Neil Robertson and Paul D. Seymour. Pathwidth sits between treewidth and vertex separation measures used by researchers at Bell Labs and in results by Michael Fellows and Rodney G. Downey. It relates to bandwidth, cutwidth, and linear layouts investigated in algorithmic graph theory from groups at Carnegie Mellon University, University of California, Berkeley, and Ecole Polytechnique. Classical inequalities connect pathwidth with tree-depth and with parameters appearing in work by Seymour', Robin Thomas, and collaborators on grid-minor theory.
Determining whether a graph has pathwidth at most k is NP-complete in general, a complexity fact paralleling hardness results from Richard Karp and followed up by reductions using constructions inspired by techniques from Stephen Cook and Leonid Levin. Fixed-parameter tractable algorithms parameterized by k have been developed by teams at University of Warwick, University of Bergen, and ETH Zurich, leveraging dynamic programming and meta-theorems related to work by Michael R. Garey and David S. Johnson. Approximation algorithms and exact exponential-time algorithms draw on methods from Nikos K. Sidiropoulos's network optimization research and practical solvers from Google Research and Microsoft Research. Heuristics for computing path decompositions are used in compilers and verification tools originating at IBM and Intel.
Structural characterizations of pathwidth involve forbidden minors and obstructions studied in the graph minor project led by Neil Robertson and Paul D. Seymour, with later contributions by Robin Thomas and others. Finite obstruction sets exist for fixed pathwidth values, paralleling the graph-minor theory that produced the Robertson–Seymour theorem and results about well-quasi-ordering. Relationships with planar graphs and grid minors tie pathwidth considerations to classical theorems by Kuratowski and later planar embeddings studied at California Institute of Technology and University of Illinois Urbana–Champaign. Decomposition theorems employed in structural graph theory reference techniques used in proofs of the Four Color Theorem and in work by researchers at Institute for Advanced Study.
Pathwidth has applications in VLSI design problems influenced by research at Bell Labs and MIT Lincoln Laboratory, in register allocation and program analysis from work at Stanford University and Princeton University, and in phylogenetic reconstruction efforts within groups at Smithsonian Institution and Natural History Museum, London. Specific graph families such as paths, trees, grids, and outerplanar graphs provide canonical examples: a path has pathwidth 1, trees have small pathwidth characterized in studies by S. L. Hakimi, and n×n grid graphs have pathwidth growing with n, connecting to combinatorial constructions by Noga Alon and László Lovász. Empirical applications include network routing research at Cisco Systems and constraint satisfaction problems explored by teams at Carnegie Mellon University.
Variants include linear-width measures like linear treewidth studied in work at ETH Zurich and University of Bonn, and directed versions considered in algorithmic graph theory papers from University of Tokyo and Seoul National University. Generalizations link to branchwidth and carving width used in graph algorithms literature by researchers at INRIA and Bell Laboratories Research, and to parameterized measures appearing in parameter ecology and complexity theory explored by Rodney G. Downey and Michael R. Fellows. Extensions to hypergraphs and matroids appear in combinatorial studies at University of British Columbia and University of Melbourne.
Category:Graph invariants