Title : Level - Treewidth Property , Exact Algorithms and Approximation Schemes

Informally, a class of graphs G is said to have the level-treewidth property (LT-property) if for every G E G there is a layout (breadth first ordering) LG such that the subgraph induced by the vertices in k-consecutive levels in the layout have treewidth O ( f ( k ) ) , for some function f . We show that several important and well known classes of graphs including planar and bounded genus graphs, (T, s)-civilized graphs, etc, satisfy the LT-property. Building on the recent work of Hunt et al. [HM+94], Khanna and Motwani [KM96] and Eppstein [Ep95], we present two general types of results for the class of graphs obeying the LT-property. 1. All problems in the classes MPSAT, TMAX and TMlN (first considered in [KM96]) have polynomial time approximation schemes. 2. The problems considered in Eppstein [Ep95] (such as subgraph isomorphism, shortest path queries, etc,) have eficient polynomial time algorithms. These results can be extended to obtain polynomial time approximation algorithms and approximation schemes for a number of PSPACE-hard combinatorial problems specified using different kinds of succinct specifications studied in [MH+95, Or82a, LW87al. Many of the results can also be extended to 6-near genus and 6-near civilized graphs, for any fixed 6. Our results significantly extend the work in [KM96, MH+95, DTS93, Ep951 and affirmatively answer recent open questions in [KM96, Ep951.