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On the solution‐space geometry of random constraint satisfaction problems
Author(s) -
Achlioptas Dimitris,
CojaOghlan Amin,
RicciTersenghi Federico
Publication year - 2011
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20323
Subject(s) - constraint satisfaction problem , random graph , mathematics , struct , satisfiability , combinatorics , constraint satisfaction , space (punctuation) , complexity of constraint satisfaction , discrete mathematics , graph coloring , random element , constraint (computer aided design) , random field , graph , local consistency , computer science , geometry , statistics , probabilistic logic , programming language , operating system
For various random constraint satisfaction problems there is a significant gap between the largest constraint density for which solutions exist and the largest density for which any polynomial time algorithm is known to find solutions. Examples of this phenomenon include random k ‐SAT, random graph coloring, and a number of other random constraint satisfaction problems. To understand this gap, we study the structure of the solution space of random k ‐SAT (i.e., the set of all satisfying assignments viewed as a subgraph of the Hamming cube). We prove that for densities well below the satisfiability threshold, the solution space decomposes into an exponential number of connected components and give quantitative bounds for the diameter, volume, and number.© 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 251–268, 2011