Search Results

For a bipartite graph G we consider the problem of finding a maximum size/weight square-free 2-matching and its generalization - the problem of finding a maximum size/weight K_{t,t}-free t-matching, where t is an integer greater than two and K_{t,t} denotes a bipartite clique with t vertices on each of the two sides. Since the weighted versions of these problems are NP-hard in general, we assume that the weights are vertex-induced on any subgraph isomorphic to K_{t,t}. We present simple combinatorial algorithms for these problems. Our algorithms are significantly simpler and faster than those previously known. We dispense with the need to shrink squares and, more generally subgraphs isomorphic to K_{t,t}, the operation which occurred in all previous algorithms for such t-matchings and instead use so-called half-edges. A half-edge of edge e is, informally speaking, a half of e containing exactly one of its endpoints. Additionally, we consider another problem concerning restricted matchings. Given a (not necessarily bipartite) graph G = (V,E), a set of k subsets of edges E₁, E₂, …, E_k and k natural numbers r₁, r₂, …, r_k, the Restricted Matching Problem asks to find a maximum size matching of G among such ones that for each 1 ≤ i ≤ k, M contains at most r_i edges of E_i. This problem is NP-hard even when G is bipartite. We show that it is solvable in polynomial time if (i) for each i the graph G contains a clique or a bipartite clique on all endpoints of E_i; in the case of a bipartite clique it is required to contain E_i and (ii) the sets E₁, …, E_k are almost vertex-disjoint - the endpoints of any two different sets have at most one vertex in common.