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Bisection of Random Cubic Graphs
[chapter]

2002
*
Lecture Notes in Computer Science
*

We present two

doi:10.1007/3-540-45726-7_10
fatcat:ybvneqiwnraxtnpg7cycmp53my
*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. ...##
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Bisecting sparse random graphs

2001
*
Random structures & algorithms (Print)
*

Consider partitions of

doi:10.1002/1098-2418(200101)18:1<31::aid-rsa3>3.3.co;2-t
fatcat:ygi3ntfuqbb5pba5q5pnio6vmu
*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

2000
*
Random structures & algorithms (Print)
*

Consider partitions of

doi:10.1002/1098-2418(200101)18:1<31::aid-rsa3>3.0.co;2-1
fatcat:fcixvpt7dbembkapxlcdd2smbi
*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

doi:10.1016/s0166-218x(98)00009-2
fatcat:7yzpcblxxzfojaymuhvrdarao4
*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

1998
*
Discrete Applied Mathematics
*

We resolve in

doi:10.1016/s0166-218x(97)00133-9
fatcat:t5hmcp5jdjgqdexpjzutk4r2gi
*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. ...##
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Linear Orderings of Random Geometric Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

Finally, we characterize

doi:10.1007/3-540-46784-x_28
fatcat:w5vh432r4bbvvnoktcfyzkzoyy
*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

2021
*
Journal of computer and system sciences (Print)
*

We show that

doi:10.1016/j.jcss.2021.02.002
fatcat:uph4ndk24nafjonawszjvy2fj4
*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. ...##
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Layout Problems on Lattice Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

This work deals with

doi:10.1007/3-540-48686-0_10
fatcat:dlte6pxzp5ebhip6nfqu3she74
*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

2009
*
Journal of Graph Algorithms and Applications
*

We show that both minimum and maximum vertex

doi:10.7155/jgaa.00179
fatcat:ntvruiu4o5a25h7pbsg4yenpx4
*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

2002
*
Information Sciences
*

These near-linear algorithms are based

doi:10.1016/s0020-0255(02)00183-4
fatcat:tou2ubel4ngo5nwqpflhqfmyjq
*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. ...##
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Approximating Layout Problems on Random Geometric Graphs

2001
*
Journal of Algorithms
*

In this paper, we study

doi:10.1006/jagm.2000.1149
fatcat:wdg35yxeaffsjdut3m27klujle
*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

2009
*
Theoretical Computer Science
*

From these two results we derive a constant expected approximation algorithm

doi:10.1016/j.tcs.2009.03.037
fatcat:tp7emumcj5eazimvwgnbgbxovm
*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]

2006
*
Lecture Notes in Computer Science
*

From these two results we derive a constant expected approximation algorithm

doi:10.1007/11940128_53
fatcat:oyo3shbv7fgyzixngxsdydc554
*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]

2020
*
arXiv
*
pre-print

We show that a simple Markov chain,

arXiv:1812.03195v4
fatcat:gbd5wvtejbfuxfgkfnyqytiuem
*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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