Covering codes and combinatorial optimization

Abstract

It was proved by Ntafos and Hakimi in 1981 (and rediscovered recently by T. Zaslavsky and the author) that cycle codes of graphs could be completely decoded in polynomial time, by reduction to the Chinese Postman problem, and use of the Edmonds and Johnson algorithm. Upper and lower bounds on the covering radius of these codes were derived by the same authors. Shortly thereafter, A. Frank proved, using matching theory that the covering radius of these codes can also be computed in polynomial time. We report on these results as well as other results of the same type concerning cocycle codes of graphs. They are dual of the former and generalize the Gale-Berlekamp switching game. We generalize the bounds on the covering radius of the cycle code of graphs to the cycle code of matroids having the sum of circuits property. This class of matroids, introduced by Seymour, contains the graphic matroids and certain cographic matroids as special cases. The associated codes can be completely decoded in polynomial time. The complexity of computing their covering radius is still unknown.