We introduce a class of digraphs analogous to proper interval graphs and bigraphs. They are defined via a geometric representation by two inclusion-free families of intervals satisfying a certain monotonicity condition; hence we call them monotone proper interval digraphs. They admit a number of equivalent definitions, including an ordering characterization by so-called MinMax orderings, and th...

In this paper, we consider the problem of nding llpreserving ordering of a sparse symmetric and positive de nite matrix such that the reordered matrix is suitable for parallel factorization. We extended the unitcost ll-preserving ordering into a generalized class that can adopt various aspects in parallel factorization, such as computation, communication and algorithmic diversity. Based on the ...

Greedy algorithms for ordering sparse matrices for Cholesky factorization can be based on diierent metrics. Minimum degree, a popular and eeective greedy ordering scheme, minimizes the number of nonzero entries in the rank-1 update (degree) at each step of the factorization. Alternatively, minimum deeciency minimizes the number of nonzero entries introduced (deeciency) at each step of the facto...

Min-Max orderings correspond to conservative lattice polymorphisms. Digraphs with Min-Max orderings have polynomial time solvable minimum cost homomorphism problems. They can also be viewed as digraph analogues of proper interval graphs and bigraphs. We give a forbidden structure characterization of digraphs with a Min-Max ordering which implies a polynomial time recognition algorithm. We also ...

Minimum Degree and Nested Dissection are the two most popular reordering schemes used to reduce ll-in and operation count when factoring and solving sparse matrices. Most of the state-of-the-art ordering packages hybridize these methods by performing incomplete Nested Dissection and ordering by Minimum Degree the subgraphs associated with the leaves of the separation tree, but to date only loos...

Ordering the columns of a nonsymmetric sparse matrix can reduce the fill created in its factorization. Minimum-degree is a popular heuristic for ordering symmetric matrices; a variant that can be used to order nonsymmetric matrices is called column minimum degree. In this paper we describe the design of a multithreaded approximate column minimum degree code. We present a framework for the algor...

When performing sparse matrix factorization, the ordering of matrix rows and columns has a dramatic impact on the factorization time. This paper describes an approach to the reordering problem that produces significantly better orderings than prior methods. The algorithm is a hybrid of nested dissection and minimum degree ordering, and combines an assortment of different algorithmic advances. N...

It has previously been shown that there exists a minimum degree ordering for regular grids that is considerably worse than nested dissection in terms of ll-in and operations for factorization 1]. This paper proves the existence of a minimum degree ordering for regular grids that has the same optimal asymptotic order complexity for ll-in and operation count as nested dissection. The analysis is ...

The use of possibilistic networks for representing conditional preference statements on discrete variables has been proposed only recently. The approach uses non-instantiated possibility weights to define conditional preference tables. Moreover, additional information about the relative strengths of these symbolic weights can be taken into account. The fact that at best we have some information...