Paper

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

ALEXANDER D. SCOTT a1 and GREGORY B. SORKIN a2
a1 Department of Mathematics, University College London, London WC1E 6BT, UK (e-mail: scott@math.ucl.ac.uk)
a2 Department of Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights NY 10598, USA (e-mail: sorkin@watson.ibm.com)

## 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.

(Revised August 7 2005)

Dedication:
For Béla Bollobás on his 60th birthday