On integer linear programs for treewidth based on perfect elimination orderings (extended version)

Many graph search algorithms use a vertex labeling to compute an ordering of the vertices. We examine such algorithms which compute a peo (perfect elimination ordering) of a chordal graph and corresponding algorithms which compute an meo (minimal elimination ordering) of a non-chordal graph, an ordering used to compute a minimal triangulation of the input graph. We express all known peo-computing search algorithms as instances of a generic algorithm called MLS (maximal label search) and generalize Algorithm MLS into CompMLS, which can compute any peo. We then extend these algorithms to versions which compute an meo and likewise generalize all known meo-computing search algorithms. We show that not all minimal triangulations can be computed by such a graph search, and, more surprisingly, that all these search algorithms compute the same set of minimal triangulations, even though the computed meos are different. Finally, we present a complexity analysis of these algorithms. An extended abstract of part of this paper was published in WG 2005 .

Cylindrical algebraic decomposition (CAD) plays an important role in field of real algebraic geometry and many other areas. As is well-known, choice of variable ordering while computing CAD has a great effect on time and memory use of computation as well as number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special kind of variable orderings (called the perfect elimination orderings''), naturally preserve chordality, which is well compatible with an important (variable) sparsity pattern called the correlative sparsity''. Experimentation suggests that if associated graph of polynomial system in question is chordal (resp., is nearly chordal), then a perfect elimination ordering of associated graph (resp., of a minimal chordal completion of associated graph) can be a good variable ordering for CAD computation. That is, by using perfect elimination orderings, CAD computation may produce a much smaller full set of projection polynomials than by using other naive variable orderings. More importantly, for complexity analysis of CAD computation via a perfect elimination ordering, an (m,d)-property of full set of projection polynomials obtained via such an ordering is given, through which size'' of this set is characterized. This property indicates that when corresponding perfect elimination tree has a lower height, full set of projection polynomials also tends to have a smaller size''. This is well consistent with experimental results, hence perfect elimination orderings with lower elimination tree height are further recommended to be used in CAD projection.

A characterization of signed graphs with generalized perfect elimination orderings

We analyze two well-known integer programming formulations for determining the treewidth of a graph that are based on perfect elimination orderings. For the first time, we prove structural properties that explain their limitations in providing convenient lower bounds and how the latter are constituted. Moreover, we investigate a flow metric approach that proved promising to achieve approximation guarantees for the pathwidth of a graph, and we show why these techniques cannot be carried over to improve the addressed formulations for the treewidth. Via computational experiments, we provide an impression on the quality and proportionality of the lower bounds on the treewidth obtained with different relaxations of perfect ordering formulations.

Generating and characterizing the perfect elimination orderings of a chordal graph

We present a new subclass of interval graphs that generalizes connected proper interval graphs. These graphs are characterized by vertex orderings called connected perfect elimination orderings (PEO), i.e., PEOs where consecutive vertices are adjacent. Alternatively, these graphs can also be characterized by special interval models and clique orderings. We present a linear-time recognition algorithm that uses PQ-trees. Furthermore, we study the behavior of multi-sweep graph searches on this graph class. This study also shows that Corneil’s well-known LBFS-recognition algorithm for proper interval graphs can be generalized to a large family of graph searches. Finally, we show that a strong result on the existence of Hamiltonian paths and cycles in proper interval graphs can be generalized to semi-proper interval graphs.

Some aspects of perfect elimination orderings in chordal graphs

Several efficient algorithms have been proposed to construct a perfect elimination ordering of the vertices of a chordal graph. We study a generalization of perfect elimination orderings, so called domination elimination orderings (deo). We show that graphs with the property that each induced subgraph has a deo (domination graphs) are related to formulas that can be reduced to formulas with a very simple structure. We also show that every brittle graph and every graph with no induced house and no chordless cycle of length at least five (HC-free graphs) are domination graphs. Moreover, every ordering produced by the Maximum Cardinality Search Procedure on an HC-free graph is a deo.KeywordsBoolean FunctionChordal GraphMaximum CardinalityPerfect GraphElimination OrderingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A recent approach for solving sparse triangular systems of equations on massively parallel computers employs a factorization of the triangular coefficient matrix to obtain a representation of its inverse in product form. The number of general communication steps required by this approach is proportional to the number of factors in the factorization. The triangular matrix can be symmetrically permuted to minimize the number of factors over suitable classes of permutations, and thereby the complexity of the parallel algorithm can be minimized. Algorithms for minimizing the number of factors over several classes of permutations have been considered in earlier work. Let $F=L+L^T$ denote the symmetric filled matrix corresponding to a Cholesky factor $L$, and let $G_F$ denote the adjacency graph of $F$. In this paper we consider the problem of minimizing the number of factors over all permutations which preserve the structure of $G_F$. The graph model of this problem is to partition the vertices $G_F$ into the fewest transitively closed subgraphs over all perfect elimination orderings while satisfying a certain precedence relationship. The solution to this chordal graph partitioning problem can be described by a greedy scheme which eliminates a largest permissible subgraph at each step. Further, the subgraph eliminated at each step can be characterized in terms of lengths of chordless paths in the current elimination graph. This solution relies on several results concerning {\em transitive perfect elimination orderings\/} introduced in this paper. We describe a partitioning algorithm with $\order{|V|+|E|}$ time and space complexity.

The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to compute a PMO (perfect moplex ordering), as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure of MLS for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph.

Vine copula structure representations using graphs and matrices

Partitioning a chordal graph into transitive subgraphs for parallel sparse triangular solution

A recent approach for solving sparse triangular systems of equations on massively parallel computers employs a factorization of the triangular coefficient matrix to obtain a representation of its inverse in product form. The number of general communication steps required by this approach is proportional to the number of factors in the factorization. The triangular matrix can be symmetrically permuted to minimize the number of factors over suitable classes of permutations, and thereby the complexity of the parallel algorithm can be minimized. Algorithms for minimizing the number of factors over several classes of permutations have been considered in earlier work. Let F = L+L[sup T] denote the symmetric filled matrix corresponding to a Cholesky factor L, and let G[sub F] denote the adjacency graph of F. In this paper we consider the problem of minirriizing the number of factors over all permutations which preserve the structure of G[sub F]. The graph model of this problem is to partition the vertices G[sub F] into the fewest transitively closed subgraphs over all perfect elimination orderings while satisfying a certain precedence relationship. The solution to this chordal graph partitioning problem can be described by a greedy scheme which eliminates a largest permissible subgraph at each step. Further, the subgraph eliminated at each step can be characterized in terms of lengths of chordless paths in the current elimination graph. This solution relies on several results concerning transitive perfect elimination orderings introduced in this paper. We describe a partitioning algorithm with [Omicron]([vert bar]V[vert bar] + [vert bar]E[vert bar]) time and space complexity.

Perfect elimination orderings of chordal powers of graphs

Chordal digraphs