OBDD-Based Representation of Interval Graphs

Abstract

A graph G = (V, E) can be described by the characteristic function of the edge set χ E which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using Ordered Binary Decision Diagrams (OBDDs) to store χ E can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e. g., quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is O(|V |/ log |V |) and the OBDD size of interval graphs is O(|V | log |V |) which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is Ω(|V | log |V |). We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using O(log |V |) operations and a coloring algorithm for unit and general intervals graphs using O(log 2 |V |) operations and evaluate the algorithms empirically. * Supported by Deutsche Forschungsgemeinschaft, grant BO 2755/1-1.

OBDD Representation of Intersection Graphs

“…The theorem implies that the ρ-OBDD size of even a path or a complete graph is Ω(N), while for some graphs, the σ-OBDD size can be o(N) as shown in [10], [11], [20] and this paper. This is the reason why we use the interleaved variable ordering.…”

“…It should be noticed that Gillé mentions in [10], [11] that the OBDD size of convex graphs is O (N log N).…”

“…In the previous works, Nunkesser and Woelfel [20], and Gillé [10], [11] have used a log N -bit label for each vertex, which is the minimal number of bits for encoding every vertices. As discussed in [4], [20], it is known to be important to keep the number of bits as low as possible by the following reason.…”

“…To prove upper bounds mentioned in the introduction, Nunkesser and Woelfel [20], and Gillé [10], [11] have used the interleaved variable ordering σ on {a n−1 , a n−2 , . .…”

“…Nunkesser and Woelfel [20] investigate the worst-case space complexities of the OBDD representations where N and M are the number of vertices and edges of a graph, respectively. Recently, Gillé [10], [11] has been improved the results as follows:…”