Nested graph dissection and approximation algorithms

This paper considers approximation algorithms for graph completion problems using the nested dissection paradigm. Given a super-additive function of interest (the smallest planar or chordal extension for example) and a test that relates it to an upper bound of the smallest separator, we provide a framework how to dissect the graph recursively such that no subgraph has more than half the value of its parent, (or is indistinguishable via separator tests) in polynomial time. Interestingly we cannot bound such a function till we have constructed the entire nested dissection. We achieve a partition of the graph with respect to a constant number of such unknown estimator functions simultaneously. Using the framework the paper presents improvements in approximating the chordal completion size (by a factor of log n), operation count (by a factor of log² n and the polynomial term depending on degree) and elimination height. We show that there exists a nested dissection ordering that simultaneously minimizes the elimination height, chordal completion, operation count to within O(log n) factors of the best possible (which may be obtained by three independent orderings) improving the previous existence theorem by factors of log n and d^1/3 log³ n for the latter two. We also show that graphs with small crossing number or fill-in have better approximations of the elimination height, completion and operation count. As a consequence we can approximate the pathwidth, cutwidth, vertex ranking problems better for such graphs. The paper also improves, in some cases, the approximation results of minimum drawing size (number of vertices plus the crossing number) of a planar embedding of a graph, and its layout area on a grid