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| Tree decomposition | |
|---|---|
| Name | Tree decomposition |
| Field | Graph theory |
| Introduced | 1986 |
| Introduced by | Robertson and Seymour |
| Related | Treewidth, Graph minor theory, Dynamic programming |
Tree decomposition is a graph-theoretic construction introduced in the work of Neil Robertson and P. D. Seymour that represents a graph by a tree whose nodes correspond to overlapping vertex subsets of the original graph. It provides a framework connecting structural graph theory in the tradition of the Graph Minor Theorem and algorithmic graph theory associated with parameters like Treewidth and notions arising in the study of Robertson–Seymour theorem. Tree decompositions underpin fixed-parameter tractable algorithms and have influenced research across combinatorics and theoretical computer science through links with celebrated results and institutions such as the ACM and the SIAM community.
A tree decomposition of a graph G = (V, E) consists of a tree T and a family of sets (bags) {B_x : x ∈ V(T)} such that for every edge of G there exists a bag containing both endpoints and for every vertex v ∈ V the set of nodes {x ∈ V(T) : v ∈ B_x} induces a connected subtree of T. The width of a tree decomposition equals max_x |B_x| − 1; the treewidth of G is the minimum width over all tree decompositions. This construction is central to the program of Neil Robertson and P. D. Seymour and appears in algorithmic results presented at venues like STOC and FOCS.
Tree decompositions capture separability and locality properties exploited in structural results such as the Graph Minor Theorem and related theorems by Robertson–Seymour. For a fixed k, the class of graphs of treewidth at most k is closed under taking minors and is subject to characterization by excluded minors, paralleling classic results by Kuratowski for planarity and influencing research connected to the Erdős–Rényi model in random graph theory. Key monotonicity properties include that adding edges cannot increase the treewidth beyond obvious bounds; minors of a graph have treewidth no larger than the original, a fact used in proofs alongside techniques from the Four Color Theorem literature and structural decompositions studied at institutions like the Institute for Advanced Study.
Important combinatorial properties relate treewidth to other invariants: bounded treewidth implies bounds on pathwidth, branchwidth, and graph genus in restricted families; conversely, families such as grids and cliques provide lower bounds and extremal examples referenced in work of Paul Erdős and László Lovász. Connections to logic and finite model theory are exemplified in applications to results credited to Moshe Vardi and the Automata theory community, where bounded treewidth yields tractability for problems expressible in monadic second-order logic, a theme celebrated in works associated with Courcelle's theorem.
Algorithms that exploit tree decompositions include dynamic programming schemes for NP-hard problems like vertex cover, graph coloring, and Hamiltonian cycle, as demonstrated in algorithmic studies published at ICALP and ESA. Exact and approximation algorithms for computing treewidth have been developed by researchers affiliated with Bell Labs, Microsoft Research, and academic groups led by figures such as Hans L. Bodlaender, whose linear-time algorithm for fixed k is a milestone in parameterized complexity published in venues like JACM. Practical solvers employ heuristics like minimum fill-in, minimum degree, and nested dissection—techniques rooted in numerical linear algebra traditions traceable to work at Argonne National Laboratory and software projects influenced by research from Stanford University and MIT.
Computational complexity results include NP-completeness of determining treewidth in general graphs, and fixed-parameter tractability results where treewidth is the parameter; these results feature prominently in courses and conferences such as IPEC and in textbooks by authors like Richard M. Karp and Michael R. Garey. Implementation and benchmarking efforts are often reported in proceedings of ALENEX and repositories maintained by research groups at University of Oslo and University of Bergen.
Tree decompositions are applied in algorithmic graph theory, constraint satisfaction problems studied in contexts like the SAT community and in research by groups at IBM Research and Google Research. They appear in bioinformatics for phylogenetic tree reconciliation problems connected to institutions such as the Wellcome Sanger Institute and in probabilistic graphical models used by researchers at Carnegie Mellon University and University of California, Berkeley for exact inference in Bayesian networks. In verification and model checking, tree decompositions facilitate algorithms for temporal logic and model checking tasks related to work at organizations like NASA and academic groups influenced by E. M. Clarke.
Other domains exploiting bounded treewidth include database theory in query evaluation studied by scholars at Princeton University and University of Edinburgh, computational linguistics in parsing algorithms developed by teams at Google and Microsoft Research, and operations research where decompositions support stochastic optimization studied at INFORMS meetings. Applications span also to network reliability analyses reported by researchers at Bellcore and to constraint optimization in artificial intelligence groups at AAAI.
Several variants generalize the tree decomposition concept: path decomposition and pathwidth relate to linear layouts studied by researchers at ETH Zurich and University of Cambridge; branch decomposition and branchwidth arise in work by A. Robertson and P. D. Seymour; clique-tree (junction tree) decompositions are central in the probabilistic graphical models literature associated with Judea Pearl and the UAI community. Other generalizations include hypertree decompositions and generalized hypertree width used in database theory and CSP research by scholars at University of Toronto and University of Florida, and matroid decompositions studied in combinatorial optimization linked to work by Jack Edmonds.
Connections extend to algebraic graph theory through tree decompositions informing matrix factorization strategies used in sparse linear algebra developed at Lawrence Livermore National Laboratory and to homological methods influenced by researchers at University of Chicago.
Canonical examples illustrate extremes: trees have treewidth 1, while cliques on n vertices have treewidth n−1; grid graphs provide families with treewidth proportional to the grid side length, a fact used in hardness reductions by researchers including László Babai and others. Constructive methods include elimination orderings corresponding to chordal completions and minimal triangulations studied by algorithm designers at Ottawa University and in literature by Hans L. Bodlaender. Practical construction heuristics such as minimum degree, nested dissection, and multilevel coarsening derive from interdisciplinary collaborations involving SIAM conferences and research centers like CNRS and Max Planck Society.