Isomorphic bisections of cubic graphs

Graph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively. In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs. Using probabilistic methods together with delicate recolouring arguments, we prove Ando's conjecture for large connected graphs

Partitioning into Isomorphic or Connected Subgraphs

This thesis deals mainly with the partitioning and connectedness of graphs. First, we show that the problem of partitioning the nodes of a graph into a specific number of subsets such that the induced subgraphs on these sets are isomorphic to one another is NP-complete. If the induced subgraphs have to be connected, the problem remains NP-complete. Then we inspect some special graph classes for which the problem is solvable in polynomial time. Afterwards, we deal with the problem of defining a polytope by incidence vectors of nodes, which induce a connected graph. We inspect some facet-defining inequalities and their general structure. For some graph classes we state the full description. We then proceed to the problem of partitioning the nodes of a graph into a given number of parts such that the induced graphs are connected. For the corresponding polytope we show the dimension and some facet defining inequalities. This theoretical inspection is advanced by the problem of partitioning a graph into different parts such that the parts induce a connected graph in order to maximize the induced cut. We introduce different ideas for solving this problem in SCIP and show the numerical results. This leads to interesting problems on MIPs in general. As the problem in literature generally deals with the feasible region, we focus on the objective function. To do that, we inspect the problem of finding MIPs for problems with nonlinear objective functions. We discuss properties and requirements showing the existence or non-existence of particular formulations. Lastly, we inspect the problem of partitioning the nodes of a graph such that all but one class are isomorphic. This problem becomes interesting if the part not inducing the isomorphism is minimized. For this purpose we also introduce a technique, which generates the parts by brute-force. Instead of partitioning the graph into isomorphic parts, we proceed to the problem of similar graphs. In this case we inspect different similarities and show algorithms which implement these

PACE: A dynamic programming algorithm for hardware/software partitioning

. This paper presents the PACE partitioning algorithm which is used in the LYCOS co-synthesis system for partitioning control/dataflow graphs into hardwareand software parts. The algorithm is a dynamic programming algorithm which solves both the problem of minimizing system execution time with a hardware area constraint and the problem of minimizing hardware area with a system execution time constraint. The target architecture consists of a single microprocessor and a single hardware chip (ASIC, FPGA, etc.) which are connected by a communication channel. The algorithm incorporates a realistic communication model and thus attempts to minimize communication overhead. The time-complexity of the algorithm is O(n 2 \Delta A) and the space-complexity is O(n \Delta A) where A is the total area of the hardware chip and n the number of code fragments which may be placed in either hardware or software. 1 Introduction The hardware/software partitioning of a system specification onto a target..

Macroscopic network circulation for planar graphs

The analysis of networks, aimed at suitably defined functionality, often focuses on partitions into subnetworks that capture desired features. Chief among the relevant concepts is a 2-partition, that underlies the classical Cheeger inequality, and highlights a constriction (bottleneck) that limits accessibility between the respective parts of the network. In a similar spirit, the purpose of the present work is to introduce a new concept of maximal global circulation and to explore 3-partitions that expose this type of macroscopic feature of networks. Herein, graph circulation is motivated by transportation networks and probabilistic flows (Markov chains) on graphs. Our goal is to quantify the large-scale imbalance of network flows and delineate key parts that mediate such global features. While we introduce and propose these notions in a general setting, in this paper, we only work out the case of planar graphs. We explain that a scalar potential can be identified to encapsulate the concept of circulation, quite similarly as in the case of the curl of planar vector fields. Beyond planar graphs, in the general case, the problem to determine global circulation remains at present a combinatorial problem

Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning

Multilevel partitioning methods that are inspired by principles of multiscaling are the most powerful practical hypergraph partitioning solvers. Hypergraph partitioning has many applications in disciplines ranging from scientific computing to data science. In this paper we introduce the concept of algebraic distance on hypergraphs and demonstrate its use as an algorithmic component in the coarsening stage of multilevel hypergraph partitioning solvers. The algebraic distance is a vertex distance measure that extends hyperedge weights for capturing the local connectivity of vertices which is critical for hypergraph coarsening schemes. The practical effectiveness of the proposed measure and corresponding coarsening scheme is demonstrated through extensive computational experiments on a diverse set of problems. Finally, we propose a benchmark of hypergraph partitioning problems to compare the quality of other solvers

A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor

Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient {\em partition oracles}. A {\em partition oracle} is a procedure that, given access to the incidence lists representation of a bounded-degree graph $G= (V,E)$ and a parameter \eps, when queried on a vertex $v\in V$, returns the part (subset of vertices) which $v$ belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most \eps |V|. In this work we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/\eps, thus improving on the result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition oracle with query complexity exponential in 1/\eps. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.Comment: 13 pages, 1 figur