Improved Algorithms for Weighted 3D-Matching

Matching problem is an important class of NP-hard problems, which has great applications in many fields, such as scheduling and code optimization etc. This paper mainly focuses on the weighted 3D-Matching problem, and gives further analysis for the structure of the problem. In order to solve the weighted 3D-Matching problem more efficiently, the weighted 3D-Matching problem is converted to the weighted 3D-Matching augmentation problem, which is to construct a maximum weighted (k+1)-matching based on a maximum weighted k-matching. The authors firstly provides further theoretical study on the structure of the weighted 3D-Matching augmentation problem and present some special properties of the problem. For the weighted 3D-Matching augmentation problem and for a given maximum weighted k-matching of the instance, there must exist two columns in this maximum weighted k-matching such that at least 2k/3 elements of those two columns are contained in the corresponding columns in maximum weighted (k+1)-matching. On the basis of those special properties and by using color-coding and dynamic programming techniques, an improved parameterized algorithm with time complexity O\+*(4.82\+[3k}) is given. The above improved algorithm results in an improved parameterized algorithm for the weighted 3D-Matching problem, which significantly improves the previous best result O\+*(5.47\+{3k}).