Matchings in convex bipartite graphs correspond to the problem of scheduling unit-length tasks on a single disjunctive resource, given a release time and a deadline for every task. The unweighted case was studied by several authors since Glover first considered the problem in 1967 [Glover, F. 1967. Maximum matching in convex bipartite graphs. Naval Res. Logist. Quart. 14 313–316] and until 1996, when Steiner and Yeomans found an algorithm whose running time is linear in the number of tasks [Steiner, G., J. S. Yeomans. 1996. A linear time algorithm for determining maximum matchings in convex, bipartite graphs. Comput. Math. Appl. 31(12) 91–96]. We address a weighted variant of this problem. Given a node-weighted convex bipartite graph G=(X, Y, E) (where Y is linearly ordered and the neighborhood of each node of X is an interval of Y), we show that it is possible to find an X-perfect matching of maximum (or minimum) weight in O(|E| + |Y| log |X|) time and O(|X| + |Y|) space. For the case in which only the nodes of Y are weighted and their weights are positive, the algorithm can be used to find a maximum-weight (not necessarily X-perfect) matching.

We present algorithms for finding a longest common increasing subsequence of two or more input sequences. For two sequences of lengths m and n, where m≥n, we present an algorithm with an output-dependent expected running time of $O((m+n\ell) \log\log \sigma + {\ensuremath{\mathit{Sort}}})$ and O(m) space, where ℓ is the length of an LCIS, σ is the size of the alphabet, and ${\ensuremath{\mathit{Sort}}}$ is the time to sort each input sequence. For k≥3 length-n sequences we present an algorithm which improves the previous best bound by more than a factor k for many inputs. In both cases, our algorithms are conceptually quite simple but rely on existing sophisticated data structures. Finally, we introduce the problem of longest common weakly-increasing (or non-decreasing) subsequences (LCWIS), for which we present an O(m+nlogn)-time algorithm for the 3-letter alphabet case. For the extensively studied longest common subsequence problem, comparable speedups have not been achieved for small alphabets.

It is shown that the problem of maintaining the topological order of the nodes of a directed acyclic graph while inserting m edges can be solved in O (min{ m 3/2 log n , m 3/2 + n 2 log n }) time, an improvement over the best known result of O ( mn ). In addition, we analyze the complexity of the same algorithm with respect to the treewidth k of the underlying undirected graph. We show that the algorithm runs in time O ( mk log 2 n ) for general k and that it can be implemented to run in O ( n log n ) time on trees, which is optimal. The algorithm also detects cycles in the input.