Fixed-parameter tractability of error correction in graphical linear systems

Paper in proceeding, 2013

In an overdetermined and feasible system of linear equations Ax=b, let vector b be corrupted, in the way that at most k entries are off their true values. Assume that we can check in the restricted system given by any minimal dependent set of rows, the correctness of all corresponding values in b. Furthermore, A has only coefficients 0 and 1, with at most two 1s in each row. We wish to recover the correct values in b and x as much as possible. The problem arises in a certain chemical mixture inference application in molecular biology, where every observable reaction product stems from at most two candidate substances. After formalization we prove that the problem is NP-hard but fixed-parameter tractable in k. The FPT result relies on the small girth of certain graphs.