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### Bisection of Random Cubic Graphs [chapter]

J. Díaz, N. Do, M. J. Sernal, N. C. Wormald
2002 Lecture Notes in Computer Science
We present two randomized algorithms to bound the bisection width of random n-vertex cubic graphs.  ...  We obtain an asymptotic upper bound for the bisection width of 0.174039n and a corresponding lower bound of 1.325961n. The analysis is based on the differential equation method.  ...  Our first result is an asymptotic bound on the bisection width of random cubic graphs. We refer the reader to [8] , for the definitions of u.a.r. (uniformly at random) and a.a.s.  ...

### Bisecting sparse random graphs

Malwina J. Luczak, Colin McDiarmid
2001 Random structures & algorithms (Print)
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  ...  We are interested in sparse random graphs Ž . G with edge probability crn.  ...  Ž . w x For each constant c ) 2, a value f c ) 0 is known 11 such that a.s. the Ž . bisection width is at least f c n, so that for each c ) 2 there is a linear lower w bound on the bisection width.  ...

### Bisecting sparse random graphs

Malwina J. Luczak, Colin McDiarmid
2000 Random structures & algorithms (Print)
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  ...  We are interested in sparse random graphs Ž . G with edge probability crn.  ...  Ž . w x For each constant c ) 2, a value f c ) 0 is known 11 such that a.s. the Ž . bisection width is at least f c n, so that for each c ) 2 there is a linear lower w bound on the bisection width.  ...

### Bisection width of transposition graphs

1998 Discrete Applied Mathematics
We prove lower and upper bounds on bisection width of transposition graphs. This class of graphs contains several frequently studied interconnection networks including star graphs and hypercubes.  ...  In particular, we prove that the bisection width of the complete transposition graph is of order O(n.n!)  ...  Corollary 4. 10 . 10 For n>3, the bisection width of the complete transposition graph satisfies bw(CT,,) d \$(n + 2)n!. H, = G be a graph on n vertices and let FH, be a component of H,,.  ...

### The Metropolis algorithm for graph bisection

Mark Jerrum, Gregory B. Sorkin
1998 Discrete Applied Mathematics
We resolve in the affirmative a question of Boppana and Bui: whether simulated annealing can. with high probability and in polynomial time, find the optimal bisection of a random graph in v.  ...  (By using a slightly modified neighborhood structure, the number of steps can bc reduced to O(n'+' ). ) We leave open the question of whether annealing is clrective for 11 in the range 1 Corresponding  ...  Acknowledgements We thank Andreas Nolte and anonymous referees for their careful reading and detailed comments.  ...

### Linear Orderings of Random Geometric Graphs [chapter]

Josep Díaz, Mathew D. Penrose, Jordi Petit, María Serna
1999 Lecture Notes in Computer Science
Finally, we characterize the probabilistic behavior of the lexicographic ordering for our layout problems on the class of random geometric graphs.  ...  We rst prove that some of these problems remain NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold with high probability on random geometric graphs.  ...  In this paper, we are concerned with bounds for several layout measures on random geometric graphs.  ...

### General Partitioning on Random Graphs

2002 Journal of Algorithms
The sparsity or density is specified by upper or lower bounds on the edge density d ∈ 0 1 , which is the fraction of actual edges present to the maximum number of edges allowed.  ...  This problem is NP-hard even for some fixed patterns and includes as special cases well-known NP-hard problems like k-coloring (each d W i = 0; each d W i W j is arbitrary), bisection (k = 2; W 1 = W 2  ...  Thus, average case results are based on models in which a bisection of small width is planted (arbitrarily or randomly) in a random graph.  ...

### Bisection of bounded treewidth graphs by convolutions

Eduard Eiben, Daniel Lokshtanov, Amer E. Mouawad
2021 Journal of computer and system sciences (Print)
We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem.  ...  On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm  ...  Algorithms for Bisection on Bounded Treewidth Graphs We start by reviewing the O(2 t+1 · n 3 )-time algorithm for solving the Bisection problem on graphs of treewidth at most t by Jansen et al.  ...

### Layout Problems on Lattice Graphs [chapter]

Josep Díaz, Mathew D. Penrose, Jordi Petit, María Serna
1999 Lecture Notes in Computer Science
This work deals with bounds on the cost of layout problems for lattice graphs and random lattice graphs.  ...  Our main result in this paper is a convergence theorem for the optimal cost of the Minimum Linear Arrangement problem and the Minimum Sum Cut problem, for the case where the underlying graph is obtained  ...  Results for the case of d-dimensional c-ary arrays (a generalization of square lattices) on the Bisection, MinCut and MinLA problems are presented in [11] .  ...

### Vertex Bisection is Hard, too

Ulrik Brandes, Daniel Fleischer
2009 Journal of Graph Algorithms and Applications
We show that both minimum and maximum vertex bisection are N P-hard, but polynomially solvable on special graph classes such as hypercubes and trees.  ...  Of eight objectives considered in that survey, only the complexity status of minimum vertex bisection is listed as unknown.  ...  For two graph classes relevant for communication networks, hypercubes and trees, the minimum vertex bisection width can either be given directly or be computed in polynomial time.  ...

### Two-dimensional packing algorithms for layout of disconnected graphs

Ugur Dogrusoz
2002 Information Sciences
These near-linear algorithms are based on strip packing, tiling, and alternate-bisection methodologies and can be used in the layout of disconnected objects in graph visualization.  ...  We present and contrast several efficient two-dimensional packing algorithms for specified aspect ratio.  ...  any overlaps to minimize the total width (height) of the bounding rectangle.  ...

### Approximating Layout Problems on Random Geometric Graphs

Josep Dı́az, Mathew D. Penrose, Jordi Petit, Marı́a Serna
2001 Journal of Algorithms
In this paper, we study the approximability of several layout problems on a family of random geometric graphs.  ...  The layout problems that we consider are: Bandwidth, Minimum Linear Arrangement, Minimum Cut Width, Minimum Sum Cut, Vertex Separation and Edge Bisection.  ...  Bounds on the cost of layout problems for lattice graphs and random lattice graphs can be found in [21] .  ...

### Balanced cut approximation in random geometric graphs

Josep Diaz, Fabrizio Grandoni, Alberto Marchetti Spaccamela
2009 Theoretical Computer Science
From these two results we derive a constant expected approximation algorithm for the β-balanced cut problem on random geometric graphs: find an edge cut of minimum size whose two sides contain at least  ...  In this paper we study the robustness of the connectivity of random geometric graphs in the supercritical phase, under deletion of edges.  ...  Acknowledgments We thank an anonymous referee for the comments that have improved the presentation of the paper.  ...

### Balanced Cut Approximation in Random Geometric Graphs [chapter]

Josep Diaz, Fabrizio Grandoni, Alberto Marchetti Spaccamela
2006 Lecture Notes in Computer Science
From these two results we derive a constant expected approximation algorithm for the β-balanced cut problem on random geometric graphs: find an edge cut of minimum size whose two sides contain at least  ...  In this paper we study the robustness of the connectivity of random geometric graphs in the supercritical phase, under deletion of edges.  ...  Acknowledgments We thank an anonymous referee for the comments that have improved the presentation of the paper.  ...

### Counting independent sets in graphs with bounded bipartite pathwidth [article]

Martin Dyer, Catherine Greenhill, Haiko Müller
2020 arXiv   pre-print
We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class.  ...  The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs.  ...  We now state our main result, which gives a bound on the mixing time of the Glauber dynamics for graphs of bounded bipartite pathwidth. Theorem 1.1.  ...
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