Results 1 
5 of
5
Engineering Planar Separator Algorithms
, 2009
"... We consider classical lineartime planar separator algorithms, determining for a given planar graph a small subset of its nodes whose removal divides the graph into two components of similar size. These algorithms are based on planar separator theorems, which guarantee separators of size O ( √ n) a ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
We consider classical lineartime planar separator algorithms, determining for a given planar graph a small subset of its nodes whose removal divides the graph into two components of similar size. These algorithms are based on planar separator theorems, which guarantee separators of size O ( √ n) and remaining components of size at most 2n/3 (where n denotes the number of nodes in the graph). In this article, we present a comprehensive experimental study of the classical algorithms applied to a large variety of graphs, where our main goal is to find separators that do not only satisfy upper bounds, but also possess other desirable characteristics with respect to separator size and component balance. We achieve this by investigating a number of specific alternatives for the concrete implementation and finetuning of certain parts of the classical algorithms. It is also shown that the choice of several parameters influences the separation quality considerably. Moreover, we propose as planar separators the usage of fundamental cycles, whose size is at most twice the diameter of the graph: For graphs of small diameter, the guaranteed bound is better than the O ( √ n) bounds, and it turns out that this simple strategy almost always outperforms the other
Breakout local search for the vertex separator problem
"... In this paper, we propose the first heuristic approach for the vertex separator problem (VSP), based on Breakout Local Search (BLS). BLS is a recent metaheuristic that follows the general framework of the popular Iterated Local Search (ILS) with a particular focus on the perturbation strategy. Base ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
In this paper, we propose the first heuristic approach for the vertex separator problem (VSP), based on Breakout Local Search (BLS). BLS is a recent metaheuristic that follows the general framework of the popular Iterated Local Search (ILS) with a particular focus on the perturbation strategy. Based on some relevant information on search history, it tries to introduce the most suitable degree of diversification by determining adaptively the number and type of moves for the next perturbation phase. The proposed heuristic is highly competitive with the exact stateofart approaches from the literature on the current VSP benchmark. Moreover, we present for the first time computational results for a set of large graphs with up to 3000 vertices, which constitutes a new challenging benchmark for VSP approaches.
A Continuous Quadratic Programming Formulation of the Vertex Separator Problem
, 2013
"... The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, 2013, doi ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, 2013, doi: 10.1007/s1010701306441), the authors announced a new continuous bilinear quadratic programming formulation of the VSP, and they used this quadratic programming problem to illustrate the new optimality conditions. The current paper develops conditions for the equivalence between this continuous quadratic program and the vertex separator problem, and it examines the relationship between the continuous formulation of the VSP and continuous quadratic programming formulations for both the edge separator problem and maximum clique problem.
Advanced Multilevel Node Separator Algorithms
, 2015
"... A node separator of a graph is a subset S of the nodes such that removing S and its incident edges divides the graph into two disconnected components of about equal size. In this work, we introduce novel algorithms to find small node separators in large graphs. With focus on solution quality, we i ..."
Abstract
 Add to MetaCart
A node separator of a graph is a subset S of the nodes such that removing S and its incident edges divides the graph into two disconnected components of about equal size. In this work, we introduce novel algorithms to find small node separators in large graphs. With focus on solution quality, we introduce novel flowbased local search algorithms which are integrated in a multilevel framework. In addition, we transfer techniques successfully used in the graph partitioning field. This includes the usage of edge ratings tailored to our problem to guide the graph coarsening algorithm as well as highly localized local search and iterated multilevel cycles to improve solution quality even further. Experiments indicate that flowbased local search algorithms on its own in a multilevel framework are already highly competitive in terms of separator quality. Adding additional local search algorithms further improves solution quality. Our strongest configuration almost always outperforms competing systems while on average computing 10 % and 62 % smaller separators than Metis and Scotch, respectively.
Continuous Optimization Continuous quadratic programming formulations of optimization problems on graphs
"... a b s t r a c t Four NPhard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program ..."
Abstract
 Add to MetaCart
(Show Context)
a b s t r a c t Four NPhard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program. This continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem, the maximum clique problem, and the maximum independent set problem. All of these formulations, when expressed as maximization problems, are shown to follow from the convexity properties of the objective function along the edges of the feasible set. An algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators. Computational results are given.