Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Planar Graph Generation with Application to Water Distribution Networks

The study of network representations of physical, biological, and social phenomena can help us better understand their structure and functional dynamics as well as formulate predictive models of these phenomena. However, in some applications there is a deficiency in real-world data-sets for research purposes due to such reasons as the data sensitivity and high costs for data retrieval. Research related to water distribution networks often relies on synthetic data because the real-world is data is not publicly available due to the sensitivity towards theft and misuse. An important characteristic of water distribution systems is that they can be embedded in a plane, therefore to simulate these system we need realistic networks which are also planar. Currently available synthetic network generators can generate networks that exhibit realism but the planarity is not guaranteed. On the other hand, existing water network generators do not guarantee similarity with the input network and do not scale. In this thesis, we present a flexible method to generate realistic water distribution networks with optimized network parameters such as pipe and tank diameters, tank minimum and maximum levels, and pump sizes. Our model consists of three stages. First, we generate a realistic planar graph from a known water network using the multi-scale randomized edit- ing. Next, we add physical water network characteristics such as pumps, pipes, tanks, and reservoirs to the obtained topology to generate a realistic synthetic water distribution system that can be used for simulation. Finally, we optimize the operational parameters using EPANet simulation tool and multi-objective optimization solver to generate a network with maximum resilience and minimum cost

Exact Algorithm for Sampling the 2D Ising Spin Glass

A sampling algorithm is presented that generates spin glass configurations of the 2D Edwards-Anderson Ising spin glass at finite temperature, with probabilities proportional to their Boltzmann weights. Such an algorithm overcomes the slow dynamics of direct simulation and can be used to study long-range correlation functions and coarse-grained dynamics. The algorithm uses a correspondence between spin configurations on a regular lattice and dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson, Proc. 8th Symp. Discrete Algorithms 258, (1997)] for sampling dimer coverings on a planar lattice is adapted to generate samplings for the dimer problem corresponding to both planar and toroidal spin glass samples. This algorithm is recursive: it computes probabilities for spins along a "separator" that divides the sample in half. Given the spins on the separator, sample configurations for the two separated halves are generated by further division and assignment. The algorithm is simplified by using Pfaffian elimination, rather than Gaussian elimination, for sampling dimer configurations. For n spins and given floating point precision, the algorithm has an asymptotic run-time of O(n^{3/2}); it is found that the required precision scales as inverse temperature and grows only slowly with system size. Sample applications and benchmarking results are presented for samples of size up to n=128^2, with fixed and periodic boundary conditions.Comment: 18 pages, 10 figures, 1 table; minor clarification

Balancing Graph Voronoi Diagrams

Abstract—Many facility location problems are concerned with minimizing operation and transportation costs by par-titioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice. We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s+1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s logn. Keywords-graph Voronoi diagram; balancing; facility loca-tion; territorial design I

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition

Electrical Flows over Spanning Trees

The network reconfiguration problem seeks to find a rooted tree $T$ such that the energy of the (unique) feasible electrical flow over $T$ is minimized. The tree requirement on the support of the flow is motivated by operational constraints in electricity distribution networks. The bulk of existing results on convex optimization over vertices of polytopes and on the structure of electrical flows do not easily give guarantees for this problem, while many heuristic methods have been developed in the power systems community as early as 1989. Our main contribution is to give the first provable approximation guarantees for the network reconfiguration problem. We provide novel lower bounds and corresponding approximation factors for various settings ranging from $\min\{O(m-n), O(n)\}$ for general graphs, to $O(\sqrt{n})$ over grids with uniform resistances on edges, and $O(1)$ for grids with uniform edge resistances and demands. To obtain the result for general graphs, we propose a new method for (approximate) spectral graph sparsification, which may be of independent interest. Using insights from our theoretical results, we propose a general heuristic for the network reconfiguration problem that is orders of magnitude faster than existing methods in the literature, while obtaining comparable performance.Comment: 37 pages, 11 figure