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Index Coding has received considerable attention recently motivated in part by real-world applications and in part by its connection to Network Coding. The basic setting of Index Coding encodes the problem input as an undirected graph and the fundamental parameter is the broadcast rate $β$, the average communication cost per bit for sufficiently long messages (i.e. the non-linear vector capacity). Recent nontrivial bounds on $β$ were derived from the study of other Index Coding capacities (e.g. the scalar capacity $β_1$) by Bar-Yossef et al (2006), Lubetzky and Stav (2007) and Alon et al (2008). However, these indirect bounds shed little light on the behavior of $β$: there was no known polynomial-time algorithm for approximating $β$ in a general network to within a nontrivial (i.e. $o(n)$) factor, and the exact value of $β$ remained unknown for any graph where Index Coding is nontrivial. Our main contribution is a direct information-theoretic analysis of the broadcast rate $β$ using linear programs, in contrast to previous approaches that compared $β$ with graph-theoretic parameters. This allows us to resolve the aforementioned two open questions. We provide a polynomial-time algorithm with a nontrivial approximation ratio for computing $β$ in a general network along with a polynomial-time decision procedure for recognizing instances with $β=2$. In addition, we pinpoint $β$ precisely for various classes of graphs (e.g. for various Cayley graphs of cyclic groups) thereby simultaneously improving the previously known upper and lower bounds for these graphs. Via this approach we construct graphs where the difference between $β$ and its trivial lower bound is linear in the number of vertices and ones where $β$ is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse.