1. Graph properties checkable in linear time in the number of vertices
- Author
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Frédéric Olive, Etienne Grandjean, System, HAL, Equipe AMACC - Laboratoire GREYC - UMR6072, Groupe de Recherche en Informatique, Image et Instrumentation de Caen (GREYC), Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN), Normandie Université (NU)-Normandie Université (NU)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN), and Normandie Université (NU)
- Subjects
Computer Networks and Communications ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,0102 computer and information sciences ,02 engineering and technology ,Finite model theory ,01 natural sciences ,Combinatorial problems ,Theoretical Computer Science ,Combinatorics ,0202 electrical engineering, electronic engineering, information engineering ,Complexity class ,Nondeterminism ,Graph property ,Time complexity ,Mathematics ,Discrete mathematics ,Complexity lower bounds ,Applied Mathematics ,Existential second-order logic ,non deterministic linear time ,Linear time ,complexity of graph properties: time complexity lower bounds ,NP ,Planarity testing ,Vertex (geometry) ,Nondeterministic algorithm ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,020201 artificial intelligence & image processing ,Feedback vertex set ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
This paper originates from the observation that many classical NP graph problems, including some NP-complete problems, are actually of very low nondeterministic time complexity. In order to formalize this observation, we define the complexity class vertexNLIN, which collects the graph problems computable on a nondeterministic RAM in time O(n), where n is the number of vertices of the input graph G=(V,E), rather than its usual size |V|+|E|. It appears that this class is robust (it is defined by a natural restrictive computational device; it is logically characterized by several simple fragments of existential second-order logic; it is closed under various combinatorial operators, including some restrictions of transitive closure) and meaningful (it contains many natural NP problems: connectivity, hamiltonicity, non-planarity, etc.). Furthermore, the very restrictive definition of vertexNLIN seems to have beneficial effects on our ability to answer difficult questions about complexity lower bounds or separation between determinism and nondeterminism. For instance, we prove that vertexNLIN strictly contains its deterministic counterpart, vertexDLIN, and even that it does not coincide with its complementary class, co-vertexNLIN. Also, we prove that several famous graph problems (e.g. planarity, 2-colourability) do not belong to vertexNLIN, although they are computable in deterministic time O(|V|+|E|).
- Published
- 2004
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