Generating Chordal Graphs with few Fill-in Edges: An Experimental Study

Graph Generation aids in analysis of graphs and their properties while insinuating conjectures via counterexamples and even generating test instances for other algorithms. In certain cases, a list of graphs will deliver numerical information for enumerative problems devoid of theoretical solutions, or even supply a source from which specimen graphs may be adopted. Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph i.e., contains no induced chordless cycle on four or more vertices. Triangulation is classified into Minimal and Minimum, where both the approach seems to minimize the number of edges added. A comparison has been drawn amongst LB-Triangulation, Lex-M and Minimum Degree Vertex (MDV) approaches to achieve triangulation with as few edges as possible along with respective run times. While LB-Triang and Lex-M algorithms provide minimal triangulation, MDV is an approximation for minimum triangulation. Determining the minimum number of edges that must be added to a bipartite graph to make it a chain graph is NP-complete. We exploit this reduction to propose a heuristic for obtaining a chain graph from a bipartite graph via a chordal graph using MDV. Dirac\u27s method of generating chordal graphs by the union is modified with the help of MDV. Nearly Chordal Graphs are generated using a novel heuristic from complete graphs. Recognition algorithm for nearly chordal graphs is introduced and the relationship between weakly chordal, nearly chordal and chordal graphs is established

Pre-processing for Triangulation of Probabilistic Networks

The currently most efficient algorithm for inference with a probabilistic network builds upon a triangulation of a network's graph. In this paper, we show that pre-processing can help in finding good triangulations forprobabilistic networks, that is, triangulations with a minimal maximum clique size. We provide a set of rules for stepwise reducing a graph, without losing optimality. This reduction allows us to solve the triangulation problem on a smaller graph. From the smaller graph's triangulation, a triangulation of the original graph is obtained by reversing the reduction steps. Our experimental results show that the graphs of some well-known real-life probabilistic networks can be triangulated optimally just by preprocessing; for other networks, huge reductions in their graph's size are obtained.Comment: Appears in Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI2001

Fast Shortest Path Distance Estimation in Large Networks

We study the problem of preprocessing a large graph so that point-to-point shortest-path queries can be answered very fast. Computing shortest paths is a well studied problem, but exact algorithms do not scale to huge graphs encountered on the web, social networks, and other applications. In this paper we focus on approximate methods for distance estimation, in particular using landmark-based distance indexing. This approach involves selecting a subset of nodes as landmarks and computing (offline) the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, we can estimate it quickly by combining the precomputed distances of the two nodes to the landmarks. We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the suggested techniques is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach in the literature which considers selecting landmarks at random. Finally, we study applications of our method in two problems arising naturally in large-scale networks, namely, social search and community detection.Yahoo! Research (internship

Drawing Planar Graphs with Few Geometric Primitives

We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with $(8n-17)/3$ segments on an $O(n)\times O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with $3n/2$ edges on an $O(n)\times O(n^2)$ grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only $(5n - 11)/3$ arcs. This is significantly smaller than the lower bound of $2n$ for line segments for a nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017