In mathematics, a graph partition is the reduction of a
graph to a smaller graph by
partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original graph, then the partitioned graph may be better suited for analysis and problem-solving than the original. Finding a partition that simplifies graph analysis is a hard problem, but one that has applications to scientific computing,
VLSI circuit design, and task scheduling in multiprocessor computers, among others.
[
] Recently, the graph partition problem has gained importance due to its application for clustering and detection of cliques in social, pathological and biological networks. For a survey on recent trends in computational methods and applications see .
Two common examples of graph partitioning are
minimum cut and
maximum cut
For a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets and , such that the number of edges between and is as large as possible. ...
problems.
Problem complexity
Typically, graph partition problems fall under the category of
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
problems. Solutions to these problems are generally derived using heuristics and approximation algorithms.
[
] However, uniform graph partitioning or a balanced graph partition problem can be shown to be
NP-complete
In computational complexity theory, a problem is NP-complete when:
# it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...
to approximate within any finite factor.
Even for special graph classes such as trees and grids, no reasonable approximation algorithms exist,
[
] unless
P=NP
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
The informal term ''quickly'', used above ...
. Grids are a particularly interesting case since they model the graphs resulting from
Finite Element Model (FEM) simulations. When not only the number of edges between the components is approximated, but also the sizes of the components, it can be shown that no reasonable fully polynomial algorithms exist for these graphs.
Problem
Consider a graph ''G'' = (''V'', ''E''), where ''V'' denotes the set of ''n'' vertices and ''E'' the set of edges. For a (''k'',''v'') balanced partition problem, the objective is to partition ''G'' into ''k'' components of at most size ''v'' · (''n''/''k''), while minimizing the capacity of the edges between separate components.
Also, given ''G'' and an integer ''k'' > 1, partition ''V'' into ''k'' parts (subsets) ''V''
1, ''V''
2, ..., ''V
k'' such that the parts are disjoint and have equal size, and the number of edges with endpoints in different parts is minimized. Such partition problems have been discussed in literature as bicriteria-approximation or resource augmentation approaches. A common extension is to
hypergraphs, where an edge can connect more than two vertices. A hyperedge is not cut if all vertices are in one partition, and cut exactly once otherwise, no matter how many vertices are on each side. This usage is common in
electronic design automation
Electronic design automation (EDA), also referred to as electronic computer-aided design (ECAD), is a category of software tools for designing electronic systems such as integrated circuits and printed circuit boards. The tools work togeth ...
.
Analysis
For a specific (''k'', 1 + ''ε'') balanced partition problem, we seek to find a minimum cost partition of ''G'' into ''k'' components with each component containing a maximum of (1 + ''ε'')·(''n''/''k'') nodes. We compare the cost of this approximation algorithm to the cost of a (''k'',1) cut, wherein each of the ''k'' components must have the same size of (''n''/''k'') nodes each, thus being a more restricted problem. Thus,
:
We already know that (2,1) cut is the minimum bisection problem and it is NP-complete.
[
] Next, we assess a 3-partition problem wherein ''n'' = 3''k'', which is also bounded in polynomial time.
Now, if we assume that we have a finite approximation algorithm for (''k'', 1)-balanced partition, then, either the 3-partition instance can be solved using the balanced (''k'',1) partition in ''G'' or it cannot be solved. If the 3-partition instance can be solved, then (''k'', 1)-balanced partitioning problem in ''G'' can be solved without cutting any edge. Otherwise, if the 3-partition instance cannot be solved, the optimum (''k'', 1)-balanced partitioning in ''G'' will cut at least one edge. An approximation algorithm with a finite approximation factor has to differentiate between these two cases. Hence, it can solve the 3-partition problem which is a contradiction under the assumption that ''P'' = ''NP''. Thus, it is evident that (''k'',1)-balanced partitioning problem has no polynomial-time approximation algorithm with a finite approximation factor unless ''P'' = ''NP''.
The
planar separator theorem states that any ''n''-vertex
planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...
can be partitioned into roughly equal parts by the removal of O() vertices. This is not a partition in the sense described above, because the partition set consists of vertices rather than edges. However, the same result also implies that every planar graph of bounded degree has a balanced cut with O() edges.
Graph partition methods
Since graph partitioning is a hard problem, practical solutions are based on heuristics. There are two broad categories of methods, local and global. Well-known local methods are the
Kernighan–Lin algorithm, and
Fiduccia-Mattheyses algorithms, which were the first effective 2-way cuts by local search strategies. Their major drawback is the arbitrary initial partitioning of the vertex set, which can affect the final solution quality. Global approaches rely on properties of the entire graph and do not rely on an arbitrary initial partition. The most common example is spectral partitioning, where a partition is derived from approximate eigenvectors of the adjacency matrix, or
spectral clustering that groups graph vertices using the
eigendecomposition of the
graph Laplacian matrix.
Multi-level methods
A multi-level graph partitioning algorithm works by applying one or more stages. Each stage reduces the size of
the graph by collapsing vertices and edges, partitions the smaller graph, then maps back and refines this partition of the original graph. A wide variety of partitioning and refinement methods can be applied within the overall multi-level scheme. In many cases, this approach can give both fast execution times and very high quality results.
One widely used example of such an approach is
METIS
Metis or Métis may refer to:
Ethnic groups
* Métis, recognized Indigenous communities in Canada and America whose distinct culture and language emerged after early intermarriage between First Nations peoples and early European settlers, primar ...
,
a graph partitioner, and hMETIS, the corresponding partitioner for hypergraphs.
[
]
An alternative approach originated from
[
]
and implemented, e.g., in
scikit-learn is
spectral clustering with the partitioning determined from
eigenvectors of the
graph Laplacian matrix for the original graph computed by
LOBPCG solver with
multigrid preconditioning.
Spectral partitioning and spectral bisection
Given a graph
with
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
, where an entry
implies an edge between node
and
, and
degree matrix , which is a diagonal matrix, where each diagonal entry of a row
,
, represents the node degree of node
. The
Laplacian matrix
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph La ...
is defined as
. Now, a ratio-cut partition for graph
is defined as a partition of
into disjoint
, and
, minimizing the ratio
:
of the number of edges that actually cross this cut to the number of pairs of vertices that could support such edges. Spectral graph partitioning can be motivated by analogy with partitioning of a vibrating string or a mass-spring system and similarly extended to the case of negative weights of the graph.
Fiedler eigenvalue and eigenvector
In such a scenario, the
second smallest eigenvalue (
) of
, yields a ''lower bound'' on the optimal cost (
) of ratio-cut partition with
. The eigenvector (
) corresponding to
, called the
''Fiedler vector'', bisects the graph into only two communities based on the ''sign of the corresponding vector entry''. Division into a larger number of communities can be achieved by repeated ''bisection'' or by using ''multiple eigenvectors'' corresponding to the smallest eigenvalues.
[
] The examples in Figures 1,2 illustrate the spectral bisection approach.
Modularity and ratio-cut
Minimum cut partitioning however fails when the number of communities to be partitioned, or the partition sizes are unknown. For instance, optimizing the cut size for free group sizes puts all vertices in the same community. Additionally, cut size may be the wrong thing to minimize since a good division is not just one with small number of edges between communities. This motivated the use of
Modularity
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
(Q)
[
] as a metric to optimize a balanced graph partition. The example in Figure 3 illustrates 2 instances of the same graph such that in ''(a)'' modularity (Q) is the partitioning metric and in ''(b)'', ratio-cut is the partitioning metric.
Applications
Conductance
Another objective function used for graph partitioning is
Conductance which is the ratio between the number of cut edges and the volume of the smallest part. Conductance is related to electrical flows and random walks. The
Cheeger bound guarantees that spectral bisection provides partitions with nearly optimal conductance. The quality of this approximation depends on the second smallest eigenvalue of the Laplacian λ
2.
Immunization
Graph partition can be useful for identifying the minimal set of nodes or links that should be immunized in order to stop epidemics.
Other graph partition methods
Spin models have been used for clustering of multivariate data wherein similarities are translated into coupling strengths.
[
] The properties of ground state spin configuration can be directly interpreted as communities. Thus, a graph is partitioned to minimize the Hamiltonian of the partitioned graph. The
Hamiltonian (H) is derived by assigning the following partition rewards and penalties.
* Reward internal edges between nodes of same group (same spin)
* Penalize missing edges in same group
* Penalize existing edges between different groups
* Reward non-links between different groups.
Additionally, Kernel-PCA-based Spectral clustering takes a form of least squares Support Vector Machine framework, and hence it becomes possible to project the data entries to a kernel induced feature space that has maximal variance, thus implying a high separation between the projected communities.
[
]
Some methods express graph partitioning as a multi-criteria optimization problem which can be solved using local methods expressed in a game theoretic framework where each node makes a decision on the partition it chooses.
For very large-scale distributed graphs classical partition methods might not apply (e.g.,
spectral partitioning, Metis
) since they require full access to graph data in order to perform global operations. For such large-scale scenarios distributed graph partitioning is used to perform partitioning through asynchronous local operations only.
Software tools
scikit-learn implements
spectral clustering with the partitioning determined from
eigenvectors of the
graph Laplacian matrix for the original graph computed by
ARPACK
ARPACK, the ARnoldi PACKage, is a numerical
software library written in FORTRAN 77 for solving large scale eigenvalue problems
in the matrix-free fashion.
The package is designed to compute a few eigenvalues and corresponding
eigenvectors of la ...
, or by
LOBPCG solver with
multigrid preconditioning.
Chaco,
[
] due to Hendrickson and Leland, implements the multilevel approach outlined above and basic local search algorithms.
Moreover, they implement spectral partitioning techniques.
METIS
Metis or Métis may refer to:
Ethnic groups
* Métis, recognized Indigenous communities in Canada and America whose distinct culture and language emerged after early intermarriage between First Nations peoples and early European settlers, primar ...
is a graph partitioning family by Karypis and Kumar. Among this family, kMetis aims at greater partitioning speed, hMetis,
applies to hypergraphs and aims at partition quality, and ParMetis
is a parallel implementation of the Metis graph partitioning algorithm.
PaToH
is another hypergraph partitioner.
KaHyPar
is a multilevel hypergraph partitioning framework providing direct k-way and recursive bisection based partitioning algorithms. It instantiates the multilevel approach in its most extreme version, removing only a single vertex in every level of the hierarchy. By using this very fine grained ''n''-level approach combined with strong local search heuristics, it computes solutions of very high quality.
Scotch
is graph partitioning framework by Pellegrini. It uses recursive multilevel bisection and includes sequential as well as parallel partitioning techniques.
Jostle
is a sequential and parallel graph partitioning solver developed by Chris Walshaw.
The commercialized version of this partitioner is known as NetWorks.
Party
implements the Bubble/shape-optimized framework and the Helpful Sets algorithm.
The software packages DibaP
and its MPI-parallel variant PDibaP
by Meyerhenke implement the Bubble framework using diffusion; DibaP also uses AMG-based techniques for coarsening and solving linear systems arising in the diffusive approach.
Sanders and Schulz released a graph partitioning package KaHIP
(Karlsruhe High Quality Partitioning) that implements for example flow-based methods, more-localized local searches and several parallel and sequential meta-heuristics.
The tools Parkway
by Trifunovic and
Knottenbelt as well as Zoltan
by Devine et al. focus on hypergraph
partitioning.
List of free open-source frameworks:
References
External links
* Chamberlain, Bradford L. (1998)
"Graph Partitioning Algorithms for Distributing Workloads of Parallel Computations"
Bibliography
*
* An exhaustive analysis of the problem from a theoretical point of view.
* One of the early fundamental works in the field. However, performance is O(n
2), so it is no longer commonly used.
* A later variant that is linear time, very commonly used, both by itself and as part of multilevel partitioning, see below.
* Multi-level partitioning is the current state of the art. This paper also has good explanations of many other methods, and comparisons of the various methods on a wide variety of problems.
* Graph partitioning (and in particular, hypergraph partitioning) has many applications to IC design.
* Simulated annealing can be used as well.
* {{cite journal , title=New spectral methods for ratio cut partitioning and clustering , last1=Hagen , first1=L. , last2=Kahng , first2=A. B. , journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems , date=September 1992 , volume=11 , issue=9 , pages= 1074–1085 , doi=10.1109/43.159993. There is a whole class of ''spectral partitioning'' methods, which use the Eigenvectors of the Laplacian of the connectivity graph. You can se
using Matlab.
NP-complete problems
Computational problems in graph theory