Perspectives of Monge properties in optimization

Bipartite permutation graphs have several nice characterizations in terms of vertex ordering. Besides, as AT-free graphs, they have a linear structure in the sense that any connected bipartite permutation graph has a dominating path. In the present paper, we elaborate the linear structure of bipartite permutation graphs by showing that any connected graph in the class can be stretched into a \"path\" with \"edges\" being chain graphs. A particular consequence from the obtained characterization is that the clique-width of bipartite permutation graphs is unbounded, which reenes a recent result of Golumbic and Rotics for permutation graphs.

On the linear structure and clique-width of bipartite permutation graphs.

On orthogonal ray graphs

Permuting matrices to avoid forbidden submatrices

We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.

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Interval Graphs: Canonical Representations in Logspace

In this paper, we further study the properties of bipartite permutation graphs. We give first efficient parallel algorithms for several problems on bipartite permutation graphs. These problems include transforming a bipartite graph into a strongly ordered one if it is also a permutation graph; testing isomorphism; finding a Hamiltonian path/cycle; solving a variant of the crossing number problem; and others. All these problems can be solved in O(log2n) time with O(n3) processors on a Common CRCW PRAM. We also show that the minimum fill-in problem for bipartite permutation graphs can be solved efficiently by a randomized parallel algorithm. © 1993 by John Wiley & Sons, Inc.

Efficient parallel algorithms for bipartite permutation graphs

Parallel recognition of the consecutive ones property with applications

Processor complexity and time complexity are two important measures of the efficiency of parallel algorithms. A problem is said to be in NC if it can be solved in polylogarithmic time using a polynomial number of processors. This dissertation centers on designing efficient parallel algorithms for discrete problems.
By exhibiting a parallel algorithm, we show, for the first time, that the recognition of the consecutive 1's property for rows of a (0,1)-matrix is in NC. The algorithm gives a useful tool for solving many other problems.
Based on the above procedure, we prove to be in NC the recognition of many classes of graphs, e.g., transformable convex bipartite (TCB) graphs, bipartite permutation (BP) graphs, $\Gamma$ circular arc graphs, $\Theta$ circular arc graphs (a.k.a. Helly circular arc graphs), proper circular arc graphs, proper interval graphs, etc. We also present the first NC algorithms for finding a maximum matching, a maximum independent set, and a minimum clique cover in a TCB graph, and for finding a Hamilton path/cycle in a BP graph. We also obtain more efficient parallel algorithms for some problems such as computing maximum cliques on comparability graphs, circle graphs, and circular arc graphs than the predecessors.
Furthermore, we study a well known longstanding open problem, the graph isomorphism problem. We introduce the notion of identification (ID) matrices. We show to be in NC the isomorphism testing for graphs which have ID matrices with the circular 1's property. These graphs include, among others, $\Gamma$ circular arc graphs, $\Theta$ circular arc graphs, and circular convex bipartite graphs. This substantially broadens the class of graphs for which the graph isomorphism problem is known to be in NC, and therefore, in P.
Included in the appendix are efficient sorting algorithms and efficient implementation of the sequential isomorphism testing algorithms, which are by no means less important.
This research has produced many efficient algorithms. They will help us in at least the following two aspects: (0) The algorithms can be used as building blocks to develop some other efficient algorithms; (1) The techniques and methods exhibited can be used to obtain efficient solutions to some other problems.

Lin Chen

Papers

Efficient parallel algorithms for bipartite permutation graphs

Parallel recognition of the consecutive ones property with applications

Parallel complexity of discrete problems