Maximum Planar Subgraph Problem Research Articles

Generating Euler Diagrams Through Combinatorial Optimization

AbstractCan a given set system be drawn as an Euler diagram? We present the first method that correctly decides this question for arbitrary set systems if the Euler diagram is required to represent each set with a single connected region. If the answer is yes, our method constructs an Euler diagram. If the answer is no, our method yields an Euler diagram for a simplified version of the set system, where a minimum number of set elements have been removed. Further, we integrate known wellformedness criteria for Euler diagrams as additional optimization objectives into our method. Our focus lies on the computation of a planar graph that is embedded in the plane to serve as the dual graph of the Euler diagram. Since even a basic version of this problem is known to be NP‐hard, we choose an approach based on integer linear programming (ILP), which allows us to compute optimal solutions with existing mathematical solvers. For this, we draw upon previous research on computing planar supports of hypergraphs and adapt existing ILP building blocks for contiguity‐constrained spatial unit allocation and the maximum planar subgraph problem. To generate Euler diagrams for large set systems, for which the proposed simplification through element removal becomes indispensable, we also present an efficient heuristic. We report on experiments with data from MovieDB and Twitter. Over all examples, including 850 non‐trivial instances, our exact optimization method failed only for one set system to find a solution without removing a set element. However, with the removal of only a few set elements, the Euler diagrams can be substantially improved with respect to our wellformedness criteria.

Properties of π-skew Graphs with Applications

The skewness of a graph G, denoted by sk(G), is the minimum number of edges in G whose removal results in a planar graph. It is an important parameter that measures how close a graph is to planarity, and it is complementary, and computationally equivalent, to the Maximum Planar Subgraph Problem. For any connected graph G on p vertices and q edges with girth g, one can easily verify that sk(G) ≥ π(G), where $$\pi(G)=\lceil q-\frac{g}{g-2}(p-2)\rceil$$ , and the graph G is said to be π-skew if equality holds. The concept of π-skew was first proposed by G. L. Chia and C. L. Lee. The π-skew graphs with girth 3 are precisely the graphs that contain a triangulation as a spanning subgraph. The purpose of this paper is to explore the properties of π-skew graphs. Some families of π-skew graphs are obtained by applying these properties, including join of two graphs, complete multipartite graphs and Cartesian product of two graphs. We also discuss the threshold for the existence of a spanning triangulation. Among other results some sufficient conditions regarding the regularity and size of a graph, which ensure a spanning triangulation, are given.

Exact Algorithms for the Maximum Planar Subgraph Problem

Given a graph G , the NP-hard Maximum Planar Subgraph problem asks for a planar subgraph of G with the maximum number of edges. The only known non-trivial exact algorithm utilizes Kuratowski’s famous planarity criterion and can be formulated as an integer linear program (ILP) or a pseudo-Boolean satisfiability problem (PBS). We examine three alternative characterizations of planarity regarding their applicability to model maximum planar subgraphs. For each, we consider both ILP and PBS variants, investigate diverse formulation aspects, and evaluate their practical performance.

Two new approximation algorithms for the maximum planar subgraph problem

The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NP-hard and it has applications in graph drawing and resource l...

New facets for the planar subgraph polytope

AbstractThis study describes certain facet classes for the planar subgraph polytope. These facets are extensions of Kuratowski facets and are of the form 2x(U) + x(E(G)\U) ≤ 2|U| + |E(G)\U| − 2 where the edge set U varies and can be empty. Two of the new types of facets complete the class of extended subdivision facets, explored by Jünger and Mutzel. In addition, the other types of facets consist of a new class of facets for the polytope called 3‐star subdivisions. It is also shown that the extended and 3‐star subdivision facets are also equivalent to members of the class of facets with coefficients in {0, 1, 2} for the set covering polytope. Computational results displaying the effectiveness of the facets in a branch‐and‐cut scheme for the maximum planar subgraph problem are presented. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

A simulated annealing algorithm for the maximum planar subgraph problem

In this paper, we introduce a new heuristic algorithm for finding maximum planar subgraphs of a graph. Our algorithm is based on the simulated annealing optimization scheme. We compare the quality of the solutions and running times of our algorithm with greedy randomized adaptive search procedure (GRASP) and branch-and-cut heuristics. We show that simulated annealing is an efficient method to find planar subgraphs with a large number of edges. The algorithm clearly outperforms earlier heuristic algorithms; it is faster and its solution quality is better. The algorithm is very simple, although it needs an implementation for the planarity test.

Skewness of Graphs with Small Cutsets

The skewness of a graph is the minimum number of edges that have to be removed to leave a planar subgraph. This is complementary, and computationally equivalent, to the Maximum Planar Subgraph problem. In this paper we look at the problem of computing the skewness of a graph with a small cutset. We show how to express the skewness of a graph with a cutset of size at most 4 in terms of the skewnesses of several derived graphs obtained by cutting along that cutset and `stitching up' afterwards. We conclude with a discussion on possible applications to planarisation.

Algorithm 797

We describe Fortran subroutines for finding approximate solutions of the maximum planar subgraph problem (graph planarization) using a Greedy Randomized Adaptive Search Procedure (GRASP). The design and implementation of the code are described in detail. Computational results with the subroutines illustrate the quality of solutions found as a function of number of GRASP iterations.

A Better Approximation Algorithm for Finding Planar Subgraphs

The MAXIMUM PLANAR SUBGRAPH problem—given a graphG, find a largest planar subgraph ofG—has applications in circuit layout, facility layout, and graph drawing. No previous polynomial-time approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1/3, which is achieved simply by producing a spanning tree ofG. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4/9 instead of 1/3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.

Maximum planar subgraphs and nice embeddings: Practical layout tools

In automatic graph drawing a given graph has to be laid out in the plane, usually according to a number of topological and aesthetic constraints. Nice drawings for sparse nonplanar graphs can be achieved by determining a maximum planar subgraph and augmenting an embedding of this graph. This approach appears to be of limited value in practice, because the maximum planar subgraph problem is NP-hard.We attack the maximum planar subgraph problem with a branch-and-cut technique which gives us quite good, and in many cases provably optimum, solutions for sparse graphs and very dense graphs. In the theoretical part of the paper, the polytope of all planar subgraphs of a graphG is defined and studied. All subgraphs of a graphG, which are subdivisions ofK 5 orK 3,3, turn out to define facets of this polytope. For cliques contained inG, the Euler inequalities turn out to be facet-defining for the planar subgraph polytope. Moreover, we introduce the subdivision inequalities,V 2k inequalities, and the flower inequalities, all of which are facet-defining for the polytope. Furthermore, the composition of inequalities by 2-sums is investigated.We also present computational experience with a branch-and-cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision ofK 5 orK 3,3. These structures give us inequalities which are used as cutting planes.Finally, we try to convince the reader that the computation of maximum planar subgraphs is indeed a practical tool for finding nice embeddings by applying this method to graphs taken from the literature.

Maximum Planar Subgraphs and Nice Embeddings: Practical Layout Tools

In automatic graph drawing a given graph has to be laid out in the plane, usually according to a number of topological and aesthetic constraints. Nice drawings for sparse nonplanar graphs can be achieved by determining a maximum planar subgraph and augmenting an embedding of this graph. This approach appears to be of limited value in practice, because the maximum planar subgraph problem is NP-hard. We attack the maximum planar subgraph problem with a branch-and-cut technique which gives us quite good, and in many cases provably optimum, solutions for sparse graphs and very dense graphs. In the theoretical part of the paper, the polytope of all planar subgraphs of a graphG is defined and studied. All subgraphs of a graphG, which are subdivisions ofK 5 orK 3,3, turn out to define facets of this polytope. For cliques contained inG, the Euler inequalities turn out to be facet-defining for the planar subgraph polytope. Moreover, we introduce the subdivision inequalities,V 2k inequalities, and the flower inequalities, all of which are facet-defining for the polytope. Furthermore, the composition of inequalities by 2-sums is investigated. We also present computational experience with a branch-and-cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision ofK 5 orK 3,3. These structures give us inequalities which are used as cutting planes. Finally, we try to convince the reader that the computation of maximum planar subgraphs is indeed a practical tool for finding nice embeddings by applying this method to graphs taken from the literature.

An $O(m\log n)$-Time Algorithm for the Maximal Planar Subgraph Problem

Based on a new version of the Hopcroft and Tarjan planarity testing algorithm, this paper develops an $O(m\log n)$-time algorithm to find a maximal planar subgraph.