  This paper presents a unified analysis framework that captures recent
advances in the study of local-optimality characterizations for codes on
graphs. These local-optimality characterizations are based on combinatorial
structures embedded in the Tanner graph of the code. Local-optimality implies
both unique maximum-likelihood (ML) optimality and unique linear-programming
(LP) decoding optimality. Also, an iterative message-passing decoding algorithm
is guaranteed to find the unique locally-optimal codeword, if one exists.
  We demonstrate this proof technique by considering a definition of
local-optimality that is based on the simplest combinatorial structures in
Tanner graphs, namely, paths of length $h$. We apply the technique of
local-optimality to a family of Tanner codes. Inverse polynomial bounds in the
code length are proved on the word error probability of LP-decoding for this
family of Tanner codes.
