A threshold for unsatisfiability
Journal of Computer and System Sciences
Gadgets, Approximation, and Linear Programming
SIAM Journal on Computing
The scaling window of the 2-SAT transition
Random Structures & Algorithms
Colouring Random Graphs in Expected Polynomial Time
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT
Discrete Applied Mathematics - The renesse issue on satisfiability
Random MAX SAT, random MAX CUT, and their phase transitions
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part II
MAX k-CUT and approximating the chromatic number of random graphs
Random Structures & Algorithms
Polynomial constraint satisfaction problems, graph bisection, and the Ising partition function
ACM Transactions on Algorithms (TALG)
The MAX-CUT of sparse random graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Linear-programming design and analysis of fast algorithms for Max 2-CSP
Discrete Optimization
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We show that a maximum cut of a random graph below the giant-component threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable constraint satisfaction problems. In addition to Max Cut, such Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max 2-Sat, Max-Ones-2-Sat, maximum independent set, and minimum vertex cover. We show that if a Max 2-CSP instance has an ‘underlying’ graph which is a random graph $G \in \mathcal{G}(n,c/n)$, then the instance is solved in linear expected time if $c \leq 1$. Moreover, for arbitrary values (or functions) $c1$ an instance is solved in expected time $n \exp(O(1+(c-1)^3 n))$; in the ‘scaling window’ $c=1+\lambda n^{-1/3}$ with $\lambda$ fixed, this expected time remains linear.Our method is to show, first, that if a Max 2-CSP has a connected underlying graph with $n$ vertices and $m$ edges, then $O(n 2^{(m-n)/2})$ is a deterministic upper bound on the solution time. Then, analysing the tails of the distribution of this quantity for a component of a random graph yields our result. Towards this end we derive some useful properties of binomial distributions and simple random walks.