# Solving sparse random instances of Max Cut and Max 2-CSP in linear expected time

## 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 ∈ script G sign(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 + λ n-1/3 with λ 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. © 2006 Cambridge University Press.