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##### This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

Addario-Berry, L. Broutin, N. and Goldschmidt, C. 2012. The continuum limit of critical random graphs. Probability Theory and Related Fields, Vol. 152, Issue. 3-4, p. 367.

Nachmias, Asaf and Peres, Yuval 2010. The critical random graph, with martingales. Israel Journal of Mathematics, Vol. 176, Issue. 1, p. 29.

Addario-Berry, L. Broutin, N. and Reed, B. 2009. Critical random graphs and the structure of a minimum spanning tree. Random Structures and Algorithms, Vol. 35, Issue. 3, p. 323.

Scott, Alexander D. and Sorkin, Gregory B. 2007. Linear-programming design and analysis of fast algorithms for Max 2-CSP. Discrete Optimization, Vol. 4, Issue. 3-4, p. 260.

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# Solving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time

• DOI: http://dx.doi.org/10.1017/S096354830500725X
• Published online: 01 January 2006
Abstract

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) $c>1$ 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.