Multilevel graph partitioning: an evolutionary approach

The graph partitioning problem is defined as that of dividing the vertices of an undirected graph into a set of balanced parts through the removal of a set of edges, whose size is to be minimized. A number of researchers have investigated multilevel schemes, which coarsen the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partitioning of the original graph. In this paper, a genetic algorithm for the coarsening phase of a multilevel scheme for graph partitioning is presented. The proposed approach has been demonstrated to improve the solution quality at the expense of running time.     

A Parallel Multilevel Metaheuristic for Graph Partitioning

One significant problem of optimisation which occurs in many scientific areas is that of graph partitioning. Several heuristics, which pertain to high quality partitions, have been put forward. Multilevel schemes can in fact improve the quality of the solutions. However, the size of the graphs is very large in many applications, making it impossible to effectively explore the search space. In these cases, parallel processing becomes a very useful tool overcoming this problem. In this paper, we propose a new parallel algorithm which uses a hybrid heuristic within a multilevel scheme. It is able to obtain very high quality partitions and improvement on those obtained by other algorithms previously put forward.     

Using domain decomposition to find graph bisectors

In this paper we introduce a three-step approach to find a vertex bisector of a graph. The first step finds adomain decomposition of the graph, consisting of a set of domains and a multisector. Eachdomain is a connected subgraph, and themultisector contains the remaining vertices that separate the domains from each other. The second step uses a block variant of the Kernighan-Lin scheme to find a bisector that is a subset of the multisector. The third step improves the bisector by bipartite graph matching. Experimental results show this domain decomposition method finds graph partitions that compare favorably with a state-of-the-art multilevel partitioning scheme in both quality and execution time. This research was supported in part by the ARPA Contract DABT63-95-C-0122, and in part by the Natural Sciences and Engineering Research Council of Canada under grant A5509.     

A Combined Evolutionary Search and Multilevel Optimisation Approach to Graph-Partitioning

The graph-partitioning problem is to divide a graph into several pieces so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges is minimised. Important applications of the problem arise, for example, in parallel processing where data sets need to be distributed across the memory of a parallel machine. Very effective heuristic algorithms have been developed for this problem which run in real-time, but it is not known how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. A distinctive feature is the use of a multilevel heuristic algorithm to provide an effective crossover. The technique is tested on several example graphs and it is demonstrated that our method can achieve extremely high quality partitions significantly better than those found by the state-of-the-art graph-partitioning packages.     

A New Method,the Fusion Fission,for the Relaxed <Emphasis Type="Italic">k</Emphasis>-way Graph Partitioning Problem,and Comparisons with Some Multilevel Algorithms

In this paper a new graph partitioning problem is introduced, the relaxed k-way graph partitioning problem. It is close to the k-way, also called multi-way, graph partitioning problem, but with relaxed imbalance constraints. This problem arises in the air traffic control area. A new graph partitioning method is presented, the Fusion Fission, which can be used to resolve the relaxed k-way graph partitioning problem. The Fusion Fission method is compared to classical Multilevel packages and with a Simulated Annealing algorithm. The Fusion Fission algorithm and the Simulated Annealing algorithm, both require a longer computation time than the Multilevel algorithms, but they also find better partitions. However, the Fusion Fission algorithm partitions the graph with a smaller imbalance and a smaller cut than Simulated Annealing does.     

Global optimization of multilevel electricity market models including network design and graph partitioning

We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in Grimm et al. (2019) that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of Grimm et al. (2019) by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.     

Bisecting sparse random graphs

Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of “cross edges” between the parts. We are interested in sparse random graphs G_{n, c/n} with edge probability c/n. We show that, if c>ln 4, then the bisection width is Ω(n) with high probability; while if c<ln 4, then it is equal to 0 with high probability. There are corresponding threshold results for partitioning into any fixed number of parts. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 31–38, 2001     

Multilevel Refinement for Combinatorial Optimisation Problems

We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.     

Multiple edges in M. Kontsevich's graph complexes and computations of the dimensions and Euler characteristics

We provide a generating function for the (graded) dimensions of M. Kontsevich's graph complexes of ordinary graphs. This generating function can be used to compute the Euler characteristic in each loop order. Furthermore, we show that graphs with multiple edges can be omitted from these graph complexes.     

Lock-Gain Based Graph Partitioning

We propose a new heuristic for the graph partitioning problem. Based on the traditional iterative improvement framework, the heuristic uses a new type of gain in selecting vertices to move between partitions. The new type of gain provides a good explanation for the performance difference of tie-breaking strategies in KL-based iterative improvement graph partitioning algorithms. The new heuristic performed excellently. Theoretical arguments supporting its efficacy are also provided. As the proposed heuristic is considered a good candidate for local optimization engines in metaheuristics, we combined it with a genetic algorithm as a sample case and obtained a surprising result that even the average results over 1,000 runs equalled the best known for most graphs.     

Judicious partitions of weighted hypergraphs

Let G be a weighted hypergraph with edges of size i for i = 1, 2. Let wi denote the total weight of edges of size i and α be the maximum weight of an edge of size 1. We study the following partitioning problem of Bollob′as and Scott: Does there exist a bipartition such that each class meets edges of total weight at least (w_1-α)/2+(2w_2)/3? We provide an optimal bound for balanced bipartition of weighted hypergraphs, partially establishing this conjecture. For dense graphs, we also give a result for partitions into more than two classes.In particular, it is shown that any graph G with m edges has a partition V_1,..., V_k such that each vertex set meets at least(1-(1-1/k)~2)m + o(m) edges, which answers a related question of Bollobás and Scott.     

On Compatible Normal Odd Partitions in Cubic Graphs

A normal odd partition of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v, we can distinguish the edge for which this vertex is pending. Three normal odd partitions are compatible whenever these distinguished edges are distinct for each vertex. We examine this notion and show that a cubic 3‐edge‐colorable graph can always be provided with three compatible normal odd partitions. The Petersen graph has this property and we can construct other cubic graphs with chromatic index four with the same property. Finally, we propose a new conjecture which, if true, would imply the well‐known Fan and Raspaud Conjecture.     

An efficient algorithm for the parallel solution of high-dimensional differential equations

The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further, we propose different heuristics for computing graph partitions tailored to adaptive waveform relaxation. We find that AWR coupled with appropriate graph partitioning methods provides a speedup by a factor between 3 and 16.     

The PI Index of Polyomino Chains of 4<Emphasis Type="Italic">k</Emphasis>-Cycles

The Padmakar-Ivan (PI) index of a graph G is the sum over all edges uv of G of the number of edges which are not equidistant from the vertices u and v. In this paper we compute the PI index of polyomino chains of 4k-cycles and establish bounds for it.     

Edge partitions of optimal 2-plane and 3-plane graphs

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal 2-plane and on optimal 3-plane graphs, which are 2-plane and 3-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple (i.e., with neither self-loops nor parallel edges) optimal 2-plane graph into a 1-plane graph and a forest, while (ii) an edge partition formed by a 1-plane graph and two plane forests always exists and can be computed in linear time. (iii) There exist efficient algorithms to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a plane graph with maximum vertex degree at most 12, or with maximum vertex degree at most 8 if the optimal2-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) There exists an infinite family of simple optimal 2-plane graphs such that in any edge partition composed of a 1-plane graph and a plane graph, the plane graph has maximum vertex degree at least 6 and the 1-plane graph has maximum vertex degree at least 12. (v) Every optimal 3-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 2-plane graph and two plane forests.     

Games induced by the partitioning of a graph

The paper aims at generalizing the notion of restricted game on a communication graph, introduced by Myerson. We consider communication graphs with weighted edges, and we define arbitrary ways of partitioning any subset of a graph, which we call correspondences. A particularly useful way to partition a graph is obtained by computing the strength of the graph. The strength of a graph is a measure introduced in graph theory to evaluate the resistance of networks under attacks, and it provides a natural partition of the graph (called the Gusfield correspondence) into resistant components. We perform a general study of the inheritance of superadditivity and convexity for the restricted game associated with a given correspondence. Our main result is to give for cycle-free graphs necessary and sufficient conditions for the inheritance of convexity of the restricted game associated with the Gusfield correspondence.     

Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems

The reformulation–linearization technique (RLT), introduced in [Sherali, H. D., Adams. W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430], provides a way to compute a hierarchy of linear programming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.     

On the validity of a front-oriented approach to partitioning large sparse graphs with a connectivity constraint

In this paper we consider the problem of partitioning large sparse graphs, such as finite element meshes. The heuristic which is proposed allows to partition into connected and quasi-balanced subgraphs in a reasonable amount of time, while attempting to minimize the number of edge cuts. Here the goal is to build partitions for graphs containing large numbers of nodes and edges, in practice at least 10⁴. Basically, the algorithm relies on the iterative construction of connected subgraphs. This construction is achieved by successively exploring clusters of nodes called fronts. Indeed, a judicious use of fronts ensures the connectivity of the subsets at low cost: it is shown that locally, i.e. for a given subgraph, the complexity of such operations grows at most linearly with the number of edges. Moreover, a few examples are given to illustrate the quality and speed of the heuristic.The work of this author was partially supported by the DGA/DRET under contract 93-1192 and by the Army Research Office under contract DAAL03-91-C-0047 (Univ. Tenn. subcontract ORA4466.04 Amendment 1).The work of this author was partially supported by the National Science Foundation under contract ASC 92-01266, the Army Research Office under contract DAAL03-91C-0047 (Univ. Tenn. subcontract ORA4466.04 Amendment 1), and ONR under contract ONR-N00014-92-J-1890.     

Fastest mixing Markov chain problem for the union of two cliques

A symmetric, random walk on a graph G can be defined by prescribing weights to the edges in such a way that for each vertex the sum of the weights of the edges incident to the vertex is at most one. The fastest mixing, Markov chain (FMMC) problem for G is to determine the weighting that yields the fastest mixing random walk. We solve the FMMC problem in the case that G is the union of two complete graphs.