We show that there is an O(n^3) algorithm to approximate the bandwidth of an AT-free graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(e + n log n) algorithm to approximate the bandwidth of an AT-free graph within a factor 4 and an O(n+ e) algorithm with a factor 6. For the special cases of permutation graphs and trapezoid graphs we obtain O(n log² n) algorithms with worst case performance ratio 2. For cocomparability graphs we obtain an O(n + e) algorithm with worst case performance ratio 3. Finally, we show that there is an O(n² log² n) algorithm to compute the exact bandwidth of chain graphs.

For an arbitrary filled graph G + of a given original graph G, we consider the problem of removing fill edges from G + in order to obtain a graph M that is both a minimal filled graph of G and a subgraph of G + . For G + with f fill edges and e original edges, we give a simple O(f(e+f)) algorithm which solves the problem and computes a corresponding minimal elimination ordering of G. We report on experiments with an implementation of our algorithm, where we test graphs G corresponding to some real sparse matrix applications and apply well-known and widely used ordering heuristics to find G + . Our findings show the amount of fill that is commonly removed by a minimalization for each of these heuristics, and also indicate that the runtime of our algorithm on these practical graphs is better than the presented worst-case bound.

We consider the following problem, called Relative Minimal Elimination Ordering. Given a graph G = (V; E) which is a subgraph of the chordal graph G 0 = (V; E 0 ), compute an inclusion minimal chordal graph G 00 = (V; E 00 ), such that E ` E 00 ` E 0 . We show that this can be done in O(nm) time. This extends the results of [2]. The algorithm is based only on well known results on chordal graphs.

. We consider the problem of removing fill edges from a filled graph G 0 to get a minimal chordal supergraph M of the original graph G; thus G ` M ` G 0 . We show that a greedy strategy can be applied if fill edges are processed for removal in the reverse order of their introduction. For a filled graph with f fill edges and e original edges, we give a simple O(f(e + f)) algorithm which solves the problem and computes a corresponding minimal elimination ordering. We believe that in practice the runtime of our algorithm is usually better than the worst-case bound of O(f(e + f)). 1 Introduction For any graph G and an ordering ff of its vertices, there is an associated set of edges called the fill that, when added to G, results in a chordal graph (G; ff). 3 The goal of finding orderings of the vertices that produce a small fill has been studied by researchers in many areas of computer science (e.g., data-base management systems [2, 14], knowledge-based systems [5, 7], computer visi...

We give an efficient NC algorithm for finding a clique separator decomposition of an arbitrary graph, that is, a series of cliques whose removal disconnects the graph. This algorithm allows one to extend a large body of results which were originally formulated for chordal graphs to other classes of graphs. Our algorithm is optimal to within a polylogarithmic factor of Tarjan's O(mn) time sequential algorithm. The decomposition can also be used to find NC algorithms for some optimization problems on special families of graphs, assuming these problems can be solved in NC for the prime graphs of the decomposition. These optimization problems include: finding a maximum-weight clique, a minimum coloring, a maximum-weight independent set, and a minimum fill-in elimination order. We also give the first parallel algorithms for solving these problems by using the clique separator decomposition. Our maximum-weight independent set algorithm applied to chordal graphs yields the most efficient know...

We prove that the problem to get an inclusion minimal elimination ordering can be solved in linear time for planar graphs. The basic structure of the linear time algorithm is as follows. We select a vertex r as maximum and get a first approximation of a minimal elimination ordering considering a vertex x as smaller than y if x has a larger distance than y from r. Using planarity, one can determine the fill-in edges joining two vertices of the same distance from r almost immediately. The algorithm determines an O(n)-representation of these fill-in edges. To determine the final fill-in ordering, we use similar techniques as in the general parallel minimal elimination algorithm of [5].

We present an alternative linear time algorithm that computes an ordering that produces a fill-in that is minimal with respect to the subset relation. It is simpler than the algorithm in [6] and is easily parallelizable. The algorithm does not rely on the computation of a breadth-first search tree.

1 Elimination Game is a well known algorithm that simulates Gaussian elimination of matrices on graphs, and it computes a triangulation of the input graph. The number of fill edges in the computed triangulation is highly dependent on the order in which Elimination Game processes the vertices, and in general the produced triangulations are neither minimum nor minimal. In order to obtain a triangulation which is close to minimum, the Minimum Degree heuristic is widely used in practice, but until now little was known on the theoretical mechanisms involved. In this paper we show some interesting properties of Elimination Game; in particular that it is able to compute a partial minimal triangulation of the input graph regardless of the order in which the vertices are processed. This results in a new algorithm to compute minimal triangulations that are sandwiched between the input graph and the triangulation resulting from Elimination Game. One of the strengths of the new approach is that is is easily parallelizable, and thus we are able to present the first parallel algorithm to compute such sandwiched minimal triangulations. In addition, the insight that we gain through Elimination Game is used to partly explain the good behavior of the Minimum Degree algorithm. We also give a new algorithm for producing minimal triangulations that is able to use the minimum degree idea to a wider extent. 1

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