Post-Processing Elimination Orderings to Reduce Induced Width

The induced width along an elimination ordering is an important factor in the space and time complexity of many inference algorithms for graphical models. Indeed, slight changes in induced width can sometimes dictate whether a particular problem is feasible (i.e. will t in memory) using variable elimination methods. For this reason, generating low width elimination orders has received extensive attention in the AI community and many heuristic ordering methods have been proposed (see e.g. [1, 2] for a nice survey). An extensive case study of several di erent heuristics was performed by Fishelson and Geiger [3]. Their results indicated that overall the Min-Fill and Weighted-Min-Fill heuristics outperformed other greedy ordering methods, including a stochastic greedy algorithm and max cardinality search. In this report, we build o of the results of Fishelson and others and ask a simple question: can post-processing be used to improve upon an already minimal induced width ordering? In particular, can we utilize the greedy edge removal method introduced in [4] to: 1) Find a minimal lled graph given the chordal graph resulting from a particular ordering; and 2) Find a better ordering given the reduced (minimal) lled graph? Choosing a low width elimination ordering is just one of several graph theory problems that involve creating a chordal supergraph from a given graph [5]. In the minimal ordering problem, the objective is to add edges ( ll-in) the graph such that the largest clique in the graph is as small as possible. In a related problem, the goal is simply to add as few edges as possible to the graph en route to making it chordal. This related problem is known as determining minimal ll. Both nding a minimum ordering and determining a minimum ll are NP-hard problems [6, 7]. Yet, as with the elimination ordering challenge, many di erent algorithms have been proposed to nd good (minimal) ll. Perhaps the best known of these algorithms is Lex-M, which is a breadth rst search algorithm that uses a lexicographic labeling scheme to identify an elimination ordering that introduces as few ll edges as possible [8]. The remainder of this report is structured as follows. In Section 2, some necessary de nitions and background are provided, including a key theorem and upon which Blair et al.'s post-processing algorithm is based. Section 3 provides a brief description of Blair et al.'s algorithm (referred to herein as MinChordal) and illustrative example. Section 4 provides some experimental results from running the post-processing algorithm on a set of UAI benchmarks. Finally, Section 5 contains some concluding remarks.