$3.($Exercise $7\mathrm{.}46$ from Kleinberg & Tardos$)$ Let G be an undirected graph where the nodes represent people and there is an edge between two nodes if the two people are friends with each other. The cohesiveness of a group S of people is defined as the ratio e$($S$)/|$S$|,$ where e$($S$)$ denotes the number of edges in S$,$ i$.$e$.,$ the number of edges that have both ends in S$.($a$)(7$ points$)$ Give a polynomial$-$time algorithm that takes a graph G and a rational num$-$ ber $\backslash$alpha $>0$ and determines whether there exists a set S whose cohesiveness is at least $\backslash$alpha $.($b$)(3$ points$)$ Give a polynomial$-$time algorithm to find a set S of nodes with maximum cohesiveness.