Average case complete problems
SIAM Journal on Computing
Structural complexity 1
On the connectivity of a random interval graph
Proceedings of the seventh international conference on Random structures and algorithms
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Algorithmic theory of random graphs
Random Structures & Algorithms - Special issue: average-case analysis of algorithms
A quantitative comparison of graph-based models for Internet topology
IEEE/ACM Transactions on Networking (TON)
Random Structures & Algorithms
First order zero-one laws for random graphs on the circle
Random Structures & Algorithms
On k-connectivity for a geometric random graph
Random Structures & Algorithms
The small-world phenomenon: an algorithmic perspective
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
A sharp concentration inequality with application
Random Structures & Algorithms
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Random 3-SAT and BDDs: The Plot Thickens Further
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
Computational Complexity and Phase Transitions
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Node-and edge-deletion NP-complete problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
On Random Intersection Graphs: The Subgraph Problem
Combinatorics, Probability and Computing
The number of neighbors needed for connectivity of wireless networks
Wireless Networks
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In combinatorial optimization problems that exhibit phase transition it is a frequently observed phenomenon that the algorithmically hard instances are concentrated around the phase transition region. The location, the size and sometimes the mere existence of this critical region, however, may depend on several factors: on the choice of an "order parameter", on the solving algorithm or on the probabilistic model. We investigate a large class of graph optimization problems and show that this concentration of hardness is in fact a more general phenomenon, if we focus on the complexity of finding or approximating the optimal value (such as the size of a maximum clique), rather than finding a witness (an actual maximum clique). Specifically, we prove that for a general class of graph optimization problems there is always a critical region of input instances in which the hardness is sharply concentrated in the following sense: (1) if the inputs that fall in the critical region are excluded, then the remaining task cannot be NP-hard, unless unlikely complexity collapses happen; (2) the critical region is a small, vanishing subset of all inputs. Thus, in this sense, the hardness of the overall task is necessarily caused by a small, exponentially vanishing critical region of the possible inputs. This concentration of hardness is invariant in the sense that it does not depend on the choice of any order parameter, or on a specific solving algorithm or on the choice of a particular probabilistic model within the considered broad family. Since a random input, drawn by any probability distribution in the family, falls almost surely outside the critical region, therefore, it is justified in a rigorous sense that the typical case complexity of these problems is easier than their worst case complexity and this phenomenon remains invariant for a broad class of models.