# An optimization involving (random) graphs

Suppose we have a graph on $n$ nodes. We would like to assign to each node either a $+1$ or a $-1$. Call this a configuration $\sigma \in \{+1,-1\}^n$. The number of $+1$s that we have to assign is exactly $s$ (hence the number of $-1$s is $n-s$.) Given a configuration $\sigma$, we look at each node $i$ and sum the values assigned to its neighbors, call this $\xi_i(\sigma)$. We then count the number of nodes for which $\xi_i(\sigma)$ is nonnegative:

$$N(\sigma) := \sum_{i=1}^n 1\{ \xi_i(\sigma) \ge 0\}.$$

The question is: what is the configuration $\sigma$ that maximizes $N(\sigma)$? Can we give a bound on $(\max N)/n$ in terms of $s/n$. If it helps, the graph can be assumed to be Erdos-Renyi.

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