Abstract
A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex uin U, the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex uin U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n^2) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.
Highlights
A bipartite graph G = (U, V, E) is convex if the vertices in V can be numbered as 1, 2, . . . , nV so that the neighbors of every vertex i ∈ U form an interval {Li, Li + 1, Li +2, . . . , Ri } ⊆ {1, 2, . . . , nV }, which we denote by [Li, Ri ], see Fig. 1a
We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n + m) time
We have the following important duality result of Yu et al [27], see [3]: Theorem 3 In a convex bipartite graph, the size of a maximum-cardinality induced matching equals the number of chain subgraphs of a minimum chain cover
Summary
The problem of finding a (classic, not induced) matching of maximum cardinality in convex bipartite graphs has been studied extensively [10,11,25] culminating in an O(n) algorithm when a compact representation of the graph is given [25]. Previous Work Yu et al [27] describe an algorithm that finds both a maximumcardinality induced matching and a minimum chain cover in a convex bipartite graph in runtime O(m2). If the input graph is not given in compact form, our algorithm can be adapted to produce a minimum chain cover (in standard representation) in O(n + m) time This improves the previous best algorithm [3], which had a runtime of O(n2).
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