Feedback vertex set

Feedback vertex set is a vertex subset problem in graph theory that asks for a set of vertices whose removal makes a graph acyclic. It arises in combinatorics, theoretical computer science, and algorithm design, and connects to many classical results and problems studied by researchers associated with institutions and conferences. The problem has deep relationships to structural graph theory, complexity theory, and parameterized algorithms.

Definition and basic properties

A feedback vertex set is defined on a finite undirected or directed graph; removing the chosen vertices yields a forest or acyclic directed graph. Fundamental properties relate to cycle structure, vertex cover, and matroidal characterizations studied by researchers at institutions such as Bell Labs, Princeton University, MIT, Stanford University, and in venues like STOC, FOCS, SODA, and ICALP. Classical theorems connect feedback vertex sets to treewidth bounds, Gallai-type decompositions, and extremal results explored by authors affiliated with University of Waterloo, University of Oxford, ETH Zurich, and University of California, Berkeley. Basic monotonicity and minimality notions mirror concepts examined in works from Cambridge University Press, Springer, and proceedings from SIAM conferences.

Complexity and algorithms

The decision version of the problem is NP-complete, a hardness result established in the lineage of reductions similar to those used in seminal papers by researchers at IBM Research, Bell Labs, and in lectures at Princeton University and Cornell University. The problem is one of the classical NP-complete problems taught alongside results involving Cook-style reductions and mentioned in textbooks from Cambridge University Press and Addison-Wesley. Exact exponential-time algorithms have been developed and improved in algorithmic papers presented at SODA, STOC, and FOCS, with contributions from research groups at Carnegie Mellon University, University of Illinois Urbana-Champaign, Microsoft Research, and Google Research. Lower bounds based on the Exponential Time Hypothesis appear in complexity studies by authors connected to ETH Zurich and University of Oxford.

Parameterized and approximation approaches

The parameterized complexity of the problem parameterized by solution size is fixed-parameter tractable, with kernelization and bounded-search-tree techniques advanced in work from groups at University of Warsaw, TU Berlin, University of Edinburgh, University of Cambridge, and KTH Royal Institute of Technology. Approximation algorithms and inapproximability thresholds have been investigated in papers from authors at Yale University, Columbia University, Harvard University, and University of Toronto, presented in venues like STOC and FOCS. Important parameterized results include linear and polynomial kernels, iterative compression methods, and randomized contraction schemes influenced by research appearing in JACM and SIAM Journal on Computing. Connections to approximation hardness often cite reductions reminiscent of classical results associated with PCP theorems and complexity frameworks developed at Rutgers University.

Special graph classes and exact results

Exact polynomial-time algorithms exist for several restricted graph classes; examples and structural characterizations have been produced by teams at University of California, Los Angeles, University of British Columbia, Tel Aviv University, Technion – Israel Institute of Technology, and Seoul National University. For planar graphs, series-parallel graphs, chordal graphs, and graphs of bounded treewidth, dedicated dynamic programming and separator-based techniques yield exact solutions; these results are discussed in conference papers at ESA, WADS, and ICALP. Combinatorial exact formulas and kernel bounds reference classical combinatorics and algorithmic graph theory literature associated with Cambridge University Press authors and monographs from Springer editors.

Applications and related problems

Applications of feedback vertex sets appear in deadlock prevention, constraint satisfaction, program analysis, and network reliability, with applied research conducted at IBM Research, Microsoft Research, Google Research, and in collaborations with industrial partners and government labs such as Los Alamos National Laboratory and Lawrence Berkeley National Laboratory. The problem is related to directed feedback arc set, vertex cover, minimum cut, and treewidth problems studied by researchers at Princeton University, Carnegie Mellon University, ETH Zurich, and Stanford University. Cross-disciplinary connections link to bioinformatics, control theory, and database theory, with interdisciplinary work presented at venues including RECOMB, CDC, and PODS.

Category:Graph theory