Graphical evolution: an introduction to the theory of random graphs
Graphical evolution: an introduction to the theory of random graphs
Many hard examples for resolution
Journal of the ACM (JACM)
The hardest constraint problems: a double phase transition
Artificial Intelligence
Experimental results on the crossover point in random 3-SAT
Artificial Intelligence - Special volume on frontiers in problem solving: phase transitions and complexity
3-coloring in time 0(1.3446^n): a no-MIS algorithm
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
A new look at the easy-hard-easy pattern of combinatorial search difficulty
Journal of Artificial Intelligence Research
The Gn,mphase transition is not hard for the Hamiltonian cycle problem
Journal of Artificial Intelligence Research
Clustering at the phase transition
AAAI'97/IAAI'97 Proceedings of the fourteenth national conference on artificial intelligence and ninth conference on Innovative applications of artificial intelligence
Evolutionary local-search with extremal optimization
Neural, Parallel & Scientific Computations
Phase Transitions and Backbones of 3-SAT and Maximum 3-SAT
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
Recognizing frozen variables in constraint satisfaction problems
Theoretical Computer Science
Preprocessing techniques for accelerating the DCOP algorithm ADOPT
Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems
Comparing two approaches to dynamic, distributed constraint satisfaction
Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems
Annals of Mathematics and Artificial Intelligence
Journal of Combinatorial Theory Series B
On the complexity of unfrozen problems
Discrete Applied Mathematics - Special issue: Typical case complexity and phase transitions
The resolution complexity of random graphk-colorability
Discrete Applied Mathematics - Special issue: Typical case complexity and phase transitions
Graph coloring in the estimation of sparse derivative matrices: Instances and applications
Discrete Applied Mathematics
Constructive generation of very hard 3-colorability instances
Discrete Applied Mathematics
Another look at graph coloring via propositional satisfiability
Discrete Applied Mathematics
Computational complexity of auditing finite attributes in statistical databases
Journal of Computer and System Sciences
Data reductions, fixed parameter tractability, and random weighted d-CNF satisfiability
Artificial Intelligence
The backdoor key: a path to understanding problem hardness
AAAI'04 Proceedings of the 19th national conference on Artifical intelligence
Phase transitions and backbones of the asymmetric traveling salesman problem
Journal of Artificial Intelligence Research
Journal of Artificial Intelligence Research
An analysis of phase transition in NK landscapes
Journal of Artificial Intelligence Research
A search space "cartography" for guiding graph coloring heuristics
Computers and Operations Research
Backbones in optimization and approximation
IJCAI'01 Proceedings of the 17th international joint conference on Artificial intelligence - Volume 1
IJCAI'01 Proceedings of the 17th international joint conference on Artificial intelligence - Volume 1
On the complexity of unfrozen problems
Discrete Applied Mathematics
The resolution complexity of random graph k-colorability
Discrete Applied Mathematics
Graph colouring heuristics guided by higher order graph properties
EvoCOP'08 Proceedings of the 8th European conference on Evolutionary computation in combinatorial optimization
On Computing Backbones of Propositional Theories
Proceedings of the 2010 conference on ECAI 2010: 19th European Conference on Artificial Intelligence
Computing infeasibility certificates for combinatorial problems through Hilbert's Nullstellensatz
Journal of Symbolic Computation
The power of ants in solving Distributed Constraint Satisfaction Problems
Applied Soft Computing
Phase transition of tractability in constraint satisfaction and bayesian network inference
UAI'03 Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence
Using hajós' construction to generate hard graph 3-colorability instances
AISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Symbolic Computation
Solving distributed CSPs using dynamic, partial centralization without explicit constraint passing
PRIMA'10 Proceedings of the 13th international conference on Principles and Practice of Multi-Agent Systems
A Study of Tabu Search for Coloring Random 3-Colorable Graphs Around the Phase Transition
International Journal of Applied Metaheuristic Computing
Average-case complexity of backtrack search for coloring sparse random graphs
Journal of Computer and System Sciences
Improving the privacy of the asynchronous partial overlay protocol
Multiagent and Grid Systems - Principles and Practice of Multi-Agent Systems
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We define the 'frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are of the same color in all legal colorings and define the collapsed graph as the one in which all frozen pairs are merged. This is analogous to studies of the development of a backbone or spine in SAT (the Satisfiability problem). We first describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3-coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs. This applies even where this probability is infinitesimal such as 10-300, although our estimates might be subject to very high variance. We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring are suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to hard test instances for graph-coloring. We present empirical evidence that the cost of coloring threshold graphs grows exponentially, when using either a specialist coloring program, or encoding into SAT, or even when using the best of both techniques. Hard instances seem to appear over an increasing range of graph connectivity as graph size increases. We give theoretical and empirical evidence to show that the size of the smallest uncolorable subgraphs of threshold graphs becomes large as the number of nodes in graphs increases. Finally, we discuss some of the issues involved in applying our work to the statistical mechanics analysis of coloring.