We consider the maximum vertex-weighted matching problem (MVM) for non-bipartite graphs in which non-negative weights are assigned to the vertices of a graph and a matching that maximizes the sum of the weights of the matched vertices is desired. In earlier work we have described a 2/3-approximation algorithm for the MVM on bipartite graphs (Florin Dobrian et al., 2019). Here we show that a 2/3-approximation algorithm for MVM on non-bipartite graphs can be obtained by restricting the length of augmenting paths to at most three. The algorithm has time complexity O(m log + n log n), where n is the number of vertices, m is the number of edges, and is the maximum degree of a vertex. The approximation ratio of the algorithm is obtained by considering failed vertices, i.e., vertices that the approximation algorithm fails to match but the exact algorithm does. We show that there are two distinct heavier matched vertices that we can charge each failed vertex to. Our proof techniques characterize the structure of augmenting paths in a novel way. We have implemented the 2/3-approximation algorithm and show that it runs in under a minute on graphs with tens of millions of vertices and hundreds of millions of edges. We compare its performance with five other algorithms: an exact algorithm for MVM, an exact algorithm for the maximum edge-weighted matching (MEM) problem, as well as three approximation algorithms. The approximation algorithms include a 1/2 approximation algorithm for MVM, and (2/3 - E)-and (1 - E)-approximation algorithms for the MEM. In our test set of nineteen problems, there are graphs on which the exact algorithms fail to terminate in 100 hours. In addition, the new 2/3-approximation algorithm for MVM outperforms the other approximation algorithms by either being faster (often by orders of magnitude) or obtaining better weights. (c) 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
In this paper, we derive generalized vertex gains for computing vertex separators, greatly improving the efficiency and data reuse in hybrid graph partitioning contexts. Using these generalized gains, we design a novel algorithm for computing vertex separators in arbitrary graphs and compare our approach to METIS, a popular graph partitioning library. In general, our hybrid algorithm scales well to very large graphs with the increased information sharing that the generalized gains afford.
We investigate various combinatorial problems arising in the minimization of the number of arithmetic operations performed during the evaluation of matrix chain products. Sparsity as well as numerical equality of arbitrary entries of the matrix factors are taken into account by the Numerical Sparse Matrix Chain Product problem which turns out to be NP-complete. The computational complexity of the Structural Sparse Matrix Chain Product problem which assumes numerically distinct matrix entries remains open. Prior work on elimination techniques for the accumulation of Jacobian matrices in the context of algorithmic differentiation is built on.
We investigate algorithms for finding column permutations of sparse matrices in order to have large diagonal entries and to have many entries symmetrically positioned around the diagonal. The aim is to improve the memory and running time requirements of a certain class of sparse direct solvers. We propose efficient algorithms for this purpose by combining two existing approaches and demonstrate the effect of our findings in practice using a direct solver. We show improvements in a number of components of the running time of a sparse direct solver with respect to the state of the art on a diverse set of matrices.
Given a large sparse Jacobian matrix with a known sparsity pattern and a positive integer p, we formulate the new combinatorial problem of maximizing the number of nonzero elements that can be discovered by forming p groups of linear combinations of columns of the matrix. This combinatorial problem is addressed by introducing a novel graph model that does not represent the underlying aspects exactly, but aims at capturing the main aspects of the situation. In an attempt to encode information on the number of discovered nonzeros, an edge-weighted column intersection graph is transformed into an edge-weighted and vertex-weighted column gain graph. This combinatorial model gives rise to heuristic algorithms which are compared by computational experiments using a set of matrices arising from different scientific disciplines.
