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

Étienne Bamas, Louis Esperet
2020 arXiv   pre-print
This paper is devoted to 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.  ... 
arXiv:1902.04899v3 fatcat:dwk5wmdigre7jahylcucmq2ywa

Parameters Fixing Strategy for Quantum Approximate Optimization Algorithm [article]

Xinwei Lee, Yoshiyuki Saito, Dongsheng Cai, Nobuyoshi Asai
2021 arXiv   pre-print
We test our strategy on 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.  ... 
arXiv:2108.05288v1 fatcat:nt5fgzpxjnewri344lkzdrcvq4

The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: Worst Case Examples [article]

Edward Farhi, David Gamarnik, Sam Gutmann
2020 arXiv   pre-print
Using this we can show that 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).  ... 
arXiv:2005.08747v1 fatcat:qahmiazlazbk3lv2ssl6y2vwme

A note on approximating Max-Bisection on regular graphs

Uriel Feige, Marek Karpinski, Michael Langberg
2001 Information Processing Letters  
We design a 0.795 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.  ... 
doi:10.1016/s0020-0190(00)00189-7 fatcat:r6qpwtuzmfc45m47igs76qxn6u

Subject Index

2007 Journal of Discrete Algorithms  
networks, 395 Dynamic programming 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  ... 
doi:10.1016/s1570-8667(07)00076-7 fatcat:wfqxglrznfb6do3wyittd5pfbi

Evaluation of QAOA based on the approximation ratio of individual samples [article]

Jason Larkin, Matías Jonsson, Daniel Justice, Gian Giacomo Guerreschi
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.  ... 
arXiv:2006.04831v2 fatcat:tphw67piqngqnbmfuqgpecktqy

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

Sergey Bravyi, Alexander Kliesch, Robert Koenig, Eugene Tang
2022 arXiv   pre-print
We show how to apply 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] .  ... 
arXiv:2011.13420v2 fatcat:njwuvyubgjgx3bi2w6utbxejwi

Hybrid quantum-classical algorithms for approximate graph coloring

Sergey Bravyi, Alexander Kliesch, Robert Koenig, Eugene Tang
2022 Quantum  
We show how to apply 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] .  ... 
doi:10.22331/q-2022-03-30-678 fatcat:2gffpxjymngnjjh3pfq6aqbb24

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

Lenka Zdeborová, Stefan Boettcher
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.  ... 
doi:10.1088/1742-5468/2010/02/p02020 fatcat:4fz6llqkbramxfbcgplut6m67i

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

Nate Veldt, David F. Gleich, Michael W. Mahoney
2016 arXiv   pre-print
An important feature 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.  ... 
arXiv:1605.08490v1 fatcat:4gb43tgz3rgtdboru7skjbnhfe

An explicit vector algorithm for high-girth MaxCut [article]

Jessica K. Thompson, Ojas Parekh, Kunal Marwaha
2021 arXiv   pre-print
We give an 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.  ... 
arXiv:2108.12477v1 fatcat:ckyu7sk5gbflxosydmzsrwlhgm

Local Improvement Gives Better Expanders [article]

Michael Lampis
2012 arXiv   pre-print
Thus, 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.  ... 
arXiv:1211.0524v1 fatcat:dwhbsvntkbd3foj2fxhp3djsn4

Survey of local algorithms

Jukka Suomela
2013 ACM Computing Surveys  
This work surveys 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] .  ... 
doi:10.1145/2431211.2431223 fatcat:zlnxn57wunh6jl5om2lagbfvim

Beyond Product State Approximations for a Quantum Analogue of Max Cut

Anurag Anshu, David Gosset, Karen Morenz, Steven T. Flammia
2020 Theory of Quantum Computation, Communication, and Cryptography  
For any instance 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  ... 
doi:10.4230/lipics.tqc.2020.7 dblp:conf/tqc/AnshuGM20 fatcat:keodgasbpre5dcrzkcrw2gwgrm
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