A New Approach to Maximum Matching in General Graphs

We reduce the problem of finding an augmenting path in a general graph to a teachability problem and show that a slight modification of depth-first search leads to an algorithm for finding such paths. As a consequence, we obtain a straightforward algorithm for maximum matching in general graphs of time complexity O(x/ffm), where n is the number of nodes and m is the number of edges in the graph. 1 I n t r o d u c t i o n a n d m o t i v a t i o n Although since Berge's theorem in 1957 [4] it is well known that for constructing a maximum matching it suffices to search for augmenting paths, until 1965 only exponential algorithms for finding a maximum matching in general graphs were known. The reason was that one did not know how to handle with odd cylces in alternating paths. In his fundamental paper [6] Edmonds solved this problem by shrinking these odd cylces. The straightforward implementation of his approach led to an O(n 4) algorithm for maximum matching in general graphs. In the suhsequence more sophisticated implementations of his approach led to O(n s) or O(nm) algorithms [2, 8, 14, 20]. In 1973 Hoperoft and Karp [12] proved the following fact. If one computes in one phase a maximal set of shortest augmenting paths, then O(vrff) such phases would be sufficient. For the bipartite case they showed that a phase can be implemented by a breath-first search followed by a depth-first search. This led to an O(n + m) implementation of one phase and hence to an O(v~m) algorithm for maximum matching in bipartite graphs. In 1975 Even and Kariv [7, 13] presented a min(u 2, m log n) implementation of a phase leading to an O(min(n 25 , v"ffmlog n)) algorithm for maximum matching in general graphs. Galil [9] called the full paper [13] "a strong contender for the ACM Longest Paper Award." Tarjan [18] called their paper "a remarkable tour-de-force." In 1978 Bartnik [3] has given an alternative O(n 2) implementation in his unpublished Ph.D. thesis (see [11]). In 1980 Mieali and Vijay Vazirani [15] claimed to have an O(m) implementation of a phase without the presentation of a proof of correctness. Although their result is cited in many papers and also in some textbooks no proof of correctness was available. Also the paper of Peterson and Loui [17] does not clarify the situation, since their paper contains only an "informally proof of correctness" which cannot be accepted as correctness proof. Possibly this shortcoming is recently repaired in [19]. The first reason, why I attempted a new approach to maximum matching in general graphs was from a didactical point of view. When I t ired lectures on algorithms for maximum matching problems I asked myself having the beautiful well-known approach for the bipartite case in mind, which first reduces the problem of finding an augmenting path to a teachability problem in a directed graph and then solves this problem by depthfirst search: Why also in the general case, we cannot reduce the problem of finding an augmenting path to a teachability problem in a directed graph and then solve this problem by something like depth-first search? I hoped then to get an approach for which it is easier to get any intuition than in the known approach. Also I hoped to get simpler algorithms. Secondly I believed that if I have sucess with this attempt, I get a straightforward O(m) implementation of a phase by something like a breath-first search followed by the new algorithm. Indeed we get simple and efficient algorithms for maximum matching in general graphs. Only the proof of correctness are more involved. In chapter 2 definitions and the general method are given. In chapter 3 the reduction to a teachability problem in a directed graph is given. This teachability problem is solved in chapter 4. The correctness proof is sketched