OBDD-Based Representation of Interval Graphs

TL;DR

This paper analyzes the size of OBDD representations for interval graphs, providing bounds and developing efficient algorithms for maximum matching and coloring, with empirical evaluation.

Contribution

It proves new bounds on OBDD sizes for interval and unit interval graphs and introduces efficient algorithms for maximum matching and coloring.

Findings

OBDD size for unit interval graphs is O(|V|/log|V|).

OBDD size for interval graphs is O(|V| log|V|).

Developed algorithms for maximum matching and coloring with logarithmic complexity.

Abstract

A graph $G = (V,E)$ can be described by the characteristic function of the edge set $\chi_E$ which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store $\chi_E$ 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 $\Omega(\ | V \ | \log \ | V \ |)$. 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.