The subject of this paper is maximum weight matchings of graphs. An edge set M of a given graph G is called a matching if and only if any pair of edges in M share no endvertices. A maximum weight matching is a matching whose total weight (total sum of edge-weights) is maximum among those of G. The maximum weight matching problem (MWM for short) is to find a maximum weight matching of a given graph. Polynomial algorithms for finding an optimum solution to MWM have already been proposed: for example, an O(|V|4) time algorithm proposed by J. Edmonds, and an O(|E||V|log |V|) time algorithm proposed by H.N. Gabow. Some applications require obtaining a matching of large total weight (not necessarily a maximum one) in realistic computing time. These existing algorithms, however, spend extremely long computing time as the size of a given graph becomes large, and several fast approximation algorithms for MWM have been proposed. In this paper, we propose six approximation algorithms GRS+, GRS_F+, GRS_R+, GRS_S+, LAM_a+ and LAM_as+. They are enhanced from known approximation ones by adding some postprocessings that consist of improved search of weight augmenting paths. Their performance is evaluated through results of computing experiment.

The minimum distance of a linear code C is a useful metric property for estimating the error correction upper bound of C and the maximum likelihood decoding of a linear code C is also of practical importance and of theoretical interest. These problems are known to be NP-hard to approximate within any constant relative error to the optimum. As a problem related to the above, we consider the maximization problem MAX-WEIGHT: Given a generator matrix of a linear code C, find a codeword c C with its weight as close to the maximum weight of C as possible. It is shown that MAX-WEIGHT PTAS unless P=NP, however, no nontrivial approximation upper and lower bounds are known. In this paper, we investigate the complexity of MAX-WEIGHT to make the approximation upper and lower bounds more precise, and show that (1) MAX-WEIGHT is APX-complete; (2) MAX-WEIGHT is approximable within relative error 1/2 to the optimum; (3) MAX-WEIGHT is not approximable within relative error 1/10 to the optimum unless P=NP.