An efficient matlab algorithm for graph partitioning

2015

In real life, there are many problems like shortest path, graph coloring, travelling Salesmen problem (TSP) etc, thus providing solution to each problem is nearly or highly impossible with the help of traditional methods in reasonable amount of time. But it may be possible with the help of heuristic approach. It provides solution but don’t guarantee optimal solution. Graph partitioning problems are NP-Complete problems, partitioning graph into p-partitions using multilevel method, spectral method etc. for various purposes .Here we are studying several techniques to partition graph.

We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.

Computational Optimization and Applications, 2012

The max-cut problem asks for partitioning the nodes V of a graph G = (V, E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NPhard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. [45] and that of Berman et al. [9]. The running time of the former can be bounded by O(|V | 3 2 log |V |). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of O([V | 3 2 (log |V |) 3 2)α(|V |), where α(|V |) is the inverse Ackermann function, it can solve large instances in practice. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time is bounded by O(|V | 3 2 log |V |), similar to the bound achieved by [45]. It can easily determine maximum cuts in huge random as well as real-world graphs with up to 10 6 nodes. We present experimental results for our method using two different matching implementations. We furthermore compare our approach with those of [45] and [9]. It turns out that our algorithm is considerably faster in practice than [45]. Moreover, it yields a much smaller associated graph. Its expanded graph size is comparable to that of [9]. However, whereas the procedure of generating the expanded graph in [9] is very involved (thus needs a sophisticated implementation), implementing our approach is an easy and straightforward task.

Corr, 2010

The c-Balanced Separator problem is a graph-partitioning problem in which given a graph G, one aims to find a cut of minimum size such that both the sides of the cut have at least cn vertices. In this paper, we present new directions of progress in the c-Balanced Separator problem. More specifically, we propose a family of mathematical programs, that depend upon a parameter p > 0, and is an extension of the uniform version of the SDPs proposed by Goemans and Linial for this problem. In fact for the case, when p = 1, if one can solve this program in polynomial time then simply using the Goemans-Williamson's randomized rounding algorithm for Max Cut [11] will give an O(1)-factor approximation algorithm for c-Balanced Separator improving the best known approximation factor of O(√ log n) due to Arora, Rao and Vazirani [4]. This family of programs is not convex but one can transform them into so called concave programs in which one optimizes a concave function over a convex feasible set. It is well known that the optima of such programs lie at one of the extreme points of the feasible set [26]. Our main contribution is a combinatorial characterization of some extreme points of the feasible set of the mathematical program, for p = 1 case, which to the best of our knowledge is the first of its kind. We further demonstrate how this characterization can be used to solve the program in a restricted setting. Non-convex programs have recently been investigated by Bhaskara and Vijayaraghvan [6] in which they design algorithms for approximating Matrix pnorms although their algorithmic techniques are analytical in nature. It is important to note that the properties of concave programs allows one to apply techniques due to Hoffman [18] or Tuy et al [26] to solve such problems with arbitrary accuracy that, for special forms of concave programs, converge in polynomial time.

Discrete Applied Mathematics

Given an undirected edge weighted graph, the graph partitioning problem consists in determining a partition of the node set of the graph into subsets of prescribed sizes, so as to maximize the sum of the weights of the edges having both endpoints in the same subset. We introduce a new class of bounds for this problem relying on the full spectral information of the weighted adjacency matrix A. The expression of these bounds involves the eigenvalues and particular geometrical parameters defined using the eigenvectors of A. A connection is established between these parameters and the maximum cut problem. We report computational results showing that the new bounds compare favorably with previous bounds in the literature.

INFORMS Journal on Computing, 2014

We derive a new semidefinite programming relaxation for the general graph partition problem (GPP). Our relaxation is based on matrix lifting with matrix variable having order equal to the number of vertices of the graph. We show that this relaxation is equivalent to the Frieze-Jerrum relaxation [A. Frieze and M. Jerrum. Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18(1):67-81, 1997] for the maximum k-cut problem with an additional constraint that involves the restrictions on the subset sizes. Since the new relaxation does not depend on the number of subsets k into which the graph should be partitioned we are able to compute bounds for large k. We compare theoretically and numerically the new relaxation with other SDP relaxations for the GPP. The results show that our relaxation provides competitive bounds and is solved significantly faster than any other known SDP bound for the general GPP.

2011

Graph partitioning is a classical graph theory problem that has proven to be NP-hard.

Journal of Global Optimization, 2010

The graph partitioning problem is to partition the vertex set of a graph into a number of nonempty subsets so that the total weight of edges connecting distinct subsets is minimized. Previous research requires the input of cardinalities of subsets or the number of subsets for equipartition. In this paper, the problem is formulated as a zero-one quadratic programming problem without the input of cardinalities. We also present three equivalent zero-one linear integer programming reformulations. Because of its importance in data biclustering, the bipartite graph partitioning is also studied. Several new methods to determine the number of subsets and the cardinalities are presented for practical applications. In addition, hierarchy partitioning and partitioning of bipartite graphs without reordering one vertex set, are studied.

2011 IEEE International Parallel & Distributed Processing Symposium, 2011

We present a novel approach to graph partitioning based on the notion of natural cuts. Our algorithm, called PUNCH, has two phases. The first phase performs a series of minimum-cut computations to identify and contract dense regions of the graph. This reduces the graph size significantly, but preserves its general structure. The second phase uses a combination of greedy and local search heuristics to assemble the final partition. The algorithm performs especially well on road networks, which have an abundance of natural cuts (such as bridges, mountain passes, and ferries). In a few minutes, it obtains the best known partitions for continental-sized networks, significantly improving on previous results.

2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence), 2008

In this paper, a new meta-method based on the physical nuclear process is presented. This meta-method called Fusion-Fission is applied to the two different class of graph partitioning problems. This paper presents results found by this method in comparison with results of classical methods for an air traffic management problem, an image segmentation problem and applied to classical benchmarks. All of these applications of the Fusion-Fission method are successful and the results found by this method outperform state-of-the-art graph partitioning packages both on classical benchmarks and on the air traffic management problem.

2021

Partitioning a connected graph into k vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph G = (V,E) and a weight function w : V → Q≥, a connected k-partition of G is a partition of V such that each class induces a connected subgraph. The balanced connected k-partition problem consists in finding a connected k-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected k-partitions that either maximize the weight of a lightest class (max-min BCPk) or minimize the weight of a heaviest class (min-max BCPk). Such problems are equivalent when k = 2, but they are different when k ≥ 3. In this work, we propose a simple pseudo-polynomial k 2 -approximation algorithm for min-max BCPk which runs in time O(W |V ||E|), where W = ∑ v∈V w(v). Based on this algorithm and using a scaling technique, we design a (polynomial) ...

Mathematical Programming, 1998

In this paper we consider the problem of k-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for cycle, cycle with ears, knapsack tree and path-block-cycle inequalities among others. The separation heuristics, plus primal heuristics, have been implemented in a branch-andcut routine using a formulation including variables for the edges with nonzero costs and node partition variables. Results are presented for three classes of problems: equipartitioning problems arising in nite element methods and partitioning problems associated with electronic circuit layout and compiler design.

Proceedings of the 30th …, 1993

HAL (Le Centre pour la Communication Scientifique Directe), 1994

Proceedings of the 1995 ACM/IEEE …, 1995