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Local approximation of the Maximum Cut in regular graphs
[article]

2020
*
arXiv
*
pre-print

This paper is devoted to

arXiv:1902.04899v3
fatcat:dwk5wmdigre7jahylcucmq2ywa
*the*distributed complexity*of*finding an*approximation**of**the**maximum**cut**in**graphs*. ... We also prove results*of*a similar flavour for*the*MAXDICUT problem*in**regular*oriented*graphs*, where we want to maximize*the*number*of*arcs oriented from*the*left part to*the*right part*of**the**cut*. ... We would like to thank Jérémie Chalopin and Keren Censor-Hillel for their remarks on*the*complexity*of*finding an orientation using very small messages*in**the*CONGEST model. ...##
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Parameters Fixing Strategy for Quantum Approximate Optimization Algorithm
[article]

2021
*
arXiv
*
pre-print

We test our strategy on

arXiv:2108.05288v1
fatcat:nt5fgzpxjnewri344lkzdrcvq4
*the*Max-*cut*problem*of*certain classes*of**graphs*such as*the*3-*regular**graphs*and*the*Erdös-Rényi*graphs*. ... However, at large circuit depth*of*QAOA, it is difficult to achieve global optimization due to*the*multiple occurrences*of**local*minima or maxima. ... There exist some classes*of**graphs*which*the*Max-*cut*problem can be solved analytically, such as*the*bipartite*graphs*,*the*2-*regular*(ring)*graphs*and*the*fullyconnected*graphs*. ...##
###
The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: Worst Case Examples
[article]

2020
*
arXiv
*
pre-print

Using this we can show that

arXiv:2005.08747v1
fatcat:qahmiazlazbk3lv2ssl6y2vwme
*the*QAOA with (d-1)^2p < n^A for any A<1, can only achieve an*approximation*ratio*of*1/2 for Max-*Cut*on bipartite random d-*regular**graphs*for d large. ... Both bipartite random d-*regular**graphs*and general random d-*regular**graphs**locally*are trees so*the*QAOA's performance is*the*same on these two ensembles. ... Examples that we focus on*in*this paper are Max-*Cut*(MC) and*Maximum*Independent Set (MIS). ...##
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A note on approximating Max-Bisection on regular graphs

2001
*
Information Processing Letters
*

We design a 0.795

doi:10.1016/s0020-0190(00)00189-7
fatcat:r6qpwtuzmfc45m47igs76qxn6u
*approximation*algorithm for*the*Max-Bisection problem restricted to*regular**graphs*.*In**the*case*of*three*regular**graphs*our results imply an*approximation*ratio*of*0.834. ... A*graph*G is said to be*regular*if all its vertices have equal degree.*In**the*following work we analyze*the*ratio between*the**Maximum*Bisection*of*any given*regular**graph*G, and its*Maximum**Cut*. ...*The*structure*of*our paper is as follows.*In*Section 2 we analyze*the*ratio between*the**Maximum*Bisection*of*any given*regular**graph*G, and its*Maximum**Cut*. ...##
###
Subject Index

2007
*
Journal of Discrete Algorithms
*

networks, 395
Dynamic programming

doi:10.1016/s1570-8667(07)00076-7
fatcat:wfqxglrznfb6do3wyittd5pfbi
*Local*similarity between quotiented ordered trees, 23; Computing*the**maximum*clique*in**the*visibility*graph**of*a simple polygon, 524;*Regular*expression constrained ... coloring problem, 533*Graph*partitioning On*the*minimum load coloring problem, 533*Graph*theory*Cut*problems*in**graphs*with a budget constraint, 262 Grid*graph**Maximum*integer multiflow and minimum ...##
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Evaluation of QAOA based on the approximation ratio of individual samples
[article]

2020
*
arXiv
*
pre-print

*In*addition, we show that

*the*QAOA performance varies significantly with

*the*

*graph*type. ... We simulate

*the*performance

*of*QAOA applied to

*the*Max-

*Cut*problem and compare it with some

*of*

*the*best classical alternatives, for exact,

*approximate*and heuristic solution. ... 4 with

*the*

*maximum*

*cut*. ...

##
###
Hybrid quantum-classical algorithms for approximate graph coloring
[article]

2022
*
arXiv
*
pre-print

We show how to apply

arXiv:2011.13420v2
fatcat:njwuvyubgjgx3bi2w6utbxejwi
*the*recursive quantum*approximate*optimization algorithm (RQAOA) to MAX-k-*CUT*,*the*problem*of*finding an*approximate*k-vertex coloring*of*a*graph*. ... First, we show that*the*standard (non-recursive) QAOA fails to solve this optimization problem for most*regular*bipartite*graphs*at any constant level p:*the**approximation*ratio achieved by QAOA is hardly ... Instead, we rely on an upper bound on*the*size*of**the**maximum*k-*cut*for typical d-*regular*random*graphs*obtained*in**the*analysis*of*an SDP-relaxation from [4] . ...##
###
Hybrid quantum-classical algorithms for approximate graph coloring

2022
*
Quantum
*

We show how to apply

doi:10.22331/q-2022-03-30-678
fatcat:2gffpxjymngnjjh3pfq6aqbb24
*the*recursive quantum*approximate*optimization algorithm (RQAOA) to MAX-k-*CUT*,*the*problem*of*finding an*approximate*k-vertex coloring*of*a*graph*. ... First, we show that*the*standard (non-recursive) QAOA fails to solve this optimization problem for most*regular*bipartite*graphs*at any constant level p:*the**approximation*ratio achieved by QAOA is hardly ... Instead, we rely on an upper bound on*the*size*of**the**maximum*k-*cut*for typical d-*regular*random*graphs*obtained*in**the*analysis*of*an SDP-relaxation from [4] . ...##
###
Page 6554 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

; Rehovot)
Improved

*approximation**of*Max-*Cut*on*graphs**of*bounded degree. ... Using computer assisted analysis, we show that for*graphs**of*maximal degree 3 our algorithm obtains an*approximation*ratio*of*at least 0.921, and for 3-*regular**graphs**the**approximation*ratio is at least ...##
###
A conjecture on the maximum cut and bisection width in random regular graphs

2010
*
Journal of Statistical Mechanics: Theory and Experiment
*

*In*this note we argue, based on theory

*of*spin glasses, that

*in*random

*regular*

*graphs*

*the*

*maximum*

*cut*size asymptotically equals

*the*number

*of*edges

*in*

*the*

*graph*minus

*the*minimum bisection size. ... Asymptotic properties

*of*random

*regular*

*graphs*are object

*of*extensive study

*in*mathematics. ... SB acknowledges support from

*the*Fulbright Commission and from

*the*US National Science Foundation through grant number DMR-0812204. ...

##
###
A Simple and Strongly-Local Flow-Based Method for Cut Improvement
[article]

2016
*
arXiv
*
pre-print

An important feature

arXiv:1605.08490v1
fatcat:4gb43tgz3rgtdboru7skjbnhfe
*of*our algorithm is that it is strongly-*local*, meaning it does not need to explore*the*entire*graph*to find*cuts*that are*locally*optimal. ... As a flow-based method, our algorithm exhibits several ad- vantages*in*terms*of**cut*optimality and accurate identification*of*target regions*in*a*graph*. ...*In*each iteration we expand*the**local**graph*, compute a small-scale*maximum*s-t flow, and then update*the**local**graph*based on this flow. ...##
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An explicit vector algorithm for high-girth MaxCut
[article]

2021
*
arXiv
*
pre-print

We give an

arXiv:2108.12477v1
fatcat:ckyu7sk5gbflxosydmzsrwlhgm
*approximation*algorithm for MaxCut and provide guarantees on*the*average fraction*of*edges*cut*on d-*regular**graphs**of*girth ≥ 2k. ... It may be viewed as a simplification*of**the*previously best known technique, which*approximates*Gaussian wave processes on*the*infinite d-*regular*tree. ...*of*Honeywell International, Inc., for*the*U.S. ...##
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Local Improvement Gives Better Expanders
[article]

2012
*
arXiv
*
pre-print

Thus,

arXiv:1211.0524v1
fatcat:dwhbsvntkbd3foj2fxhp3djsn4
*in**the*end we obtain an improvement not only for some small special cases but on*the*general asymptotic bound on*the*expansion*of*Δ-*regular**graphs*given by Bollobás. ... Thus, after again applying*the*union bound, we obtain improved lower bounds on*the*expansion*of*random Δ-*regular**graphs*for Δ> 4. ... Acknowledgement: I am grateful to Johan Håstad who first pointed out to me*the*idea*of*looking for a*locally*optimal set*of*vertices, which is*the*basis*of*this paper. ...##
###
Survey of local algorithms

2013
*
ACM Computing Surveys
*

This work surveys

doi:10.1145/2431211.2431223
fatcat:zlnxn57wunh6jl5om2lagbfvim
*the*state-*of*-*the*-art*in**the*field, covering impossibility results, deterministic*local*algorithms, randomised*local*algorithms, and*local*algorithms for geometric*graphs*. ... When we study*local*algorithms, we assume that*the*problem instance is given*in*a distributed manner: each node*in**the*communication*graph*G knows part*of**the*input. ... Acknowledgements This is*the*author's version*of**the*work;*the*definitive version will be published*in*ACM Computing Surveys [136] . ...##
###
Beyond Product State Approximations for a Quantum Analogue of Max Cut

2020
*
Theory of Quantum Computation, Communication, and Cryptography
*

For any instance

doi:10.4230/lipics.tqc.2020.7
dblp:conf/tqc/AnshuGM20
fatcat:keodgasbpre5dcrzkcrw2gwgrm
*of*this problem*the**maximum*energy attained by a product state is lower bounded by*the*Max*Cut**of**the**graph*and upper bounded by*the*standard Goemans-Williamson semidefinite programming ... We consider a computational problem where*the*goal is to*approximate**the**maximum*eigenvalue*of*a two-*local*Hamiltonian that describes Heisenberg interactions between qubits located at*the*vertices*of*a ... Recall that*the**maximum**cut**of*a weighted*graph*G is defined to be MC(G) = max z∈{±1} n*Cut*G (z) where*Cut*G (z) = {i,j}∈E w ij 2 (I − z i z j ). (2) An*approximation*algorithm for*the*Max*Cut*problem ...
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