Vertex coloring of graphs via phase dynamics of coupled oscillatory networks

While Boolean logic has been the backbone of digital information processing, there exist classes of computationally hard problems wherein this paradigm is fundamentally inefficient. Vertex coloring of graphs, belonging to the class of combinatorial optimization, represents one such problem. It is well studied for its applications in data sciences, life sciences, social sciences and technology, and hence, motivates alternate, more efficient non-Boolean pathways towards its solution. Here we demonstrate a coupled relaxation oscillator based dynamical system that exploits insulator-metal transition in Vanadium Dioxide (VO2) to efficiently solve vertex coloring of graphs. Pairwise coupled VO2 oscillator circuits have been analyzed before for basic computing operations, but using complex networks of VO2 oscillators, or any other oscillators, for more complex tasks have been challenging in theory as well as in experiments. The proposed VO2 oscillator network harnesses the natural analogue between optimization problems and energy minimization processes in highly parallel, interconnected dynamical systems to approximate optimal coloring of graphs. We further indicate a fundamental connection between spectral properties of linear dynamical systems and spectral algorithms for graph coloring. Our work not only elucidates a physics-based computing approach but also presents tantalizing opportunities for building customized analog co-processors for solving hard problems efficiently.

[1]  Chia-Lun Hu.  Self-sustained oscillation in an R_H - C or R_H - L circuit containing a hysteresis resistor R_H , 1986 .

[2]  V. Zunjar,et al.  Antioxidant activities and phenolics profiling of different parts of Carica papaya by LCMS-MS , 2015, Natural product research.

[3]  Leon O. Chua,et al.  Cellular neural networks: applications , 1988 .

[4]  Yao-Nan Wang,et al.  New mixed broadcast scheduling approach using neural networks and graph coloring in wireless sensor network , 2009 .

[5]  Narayanan Vijaykrishnan,et al.  Video analytics using beyond CMOS devices , 2014, 2014 Design, Automation & Test in Europe Conference & Exhibition (DATE).

[6]  Andrew Lucas,et al.  Ising formulations of many NP problems , 2013, Front. Physics.

[7]  N. Kopell,et al.  Anti-phase solutions in relaxation oscillators coupled through excitatory interactions , 1995, Journal of mathematical biology.

[8]  R Stanley Williams,et al.  Sub-100 fJ and sub-nanosecond thermally driven threshold switching in niobium oxide crosspoint nanodevices , 2012, Nanotechnology.

[9]  Eugene M. Izhikevich,et al.  Weakly pulse-coupled oscillators, FM interactions, synchronization, and oscillatory associative memory , 1999, IEEE Trans. Neural Networks.

[10]  Vijaykrishnan Narayanan,et al.  Synchronized charge oscillations in correlated electron systems , 2014, Scientific Reports.

[11]  Kang Li-shan,et al.  Some massively parallel algorithms from nature , 2002, Wuhan University Journal of Natural Sciences.

[12]  H T Siegelmann,et al.  Dating and Context of Three Middle Stone Age Sites with Bone Points in the Upper Semliki Valley, Zaire , 2007 .

[13]  Chandler Davis.  The rotation of eigenvectors by a perturbation , 1963 .

[14]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[15]  Hoppensteadt,et al.  Synchronization of laser oscillators, associative memory, and optical neurocomputing , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Romulus Dan Vlasie.  Systematic generation of very hard cases for graph 3-colorability , 1995, Proceedings of 7th IEEE International Conference on Tools with Artificial Intelligence.

[17]  J. Gilbert,et al.  Graph Coloring Using Eigenvalue Decomposition , 1983 .

[18]  Ivan K Schuller,et al.  Role of thermal heating on the voltage induced insulator-metal transition in VO2. , 2013, Physical review letters.

[19]  Paul M. Solomon,et al.  In Quest of the “Next Switch”: Prospects for Greatly Reduced Power Dissipation in a Successor to the Silicon Field-Effect Transistor , 2010, Proceedings of the IEEE.

[20]  Zoltán Toroczkai,et al.  Optimization hardness as transient chaos in an analog approach to constraint satisfaction , 2011, ArXiv.

[21]  D. Welsh,et al.  A Spectral Technique for Coloring Random 3-Colorable Graphs , 1994 .

[22]  Frank McSherry,et al.  Spectral partitioning of random graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[23]  Tai-Kuo Woo,et al.  Resource allocation in a dynamically partitionable bus network using a graph coloring algorithm , 1991, IEEE Trans. Commun..

[24]  Daniel Brélaz,et al.  New methods to color the vertices of a graph , 1979, CACM.

[25]  Fabrizio Bonani,et al.  Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states , 2014, Science Advances.

[26]  Nicolas Zufferey,et al.  Graph colouring approaches for a satellite range scheduling problem , 2008, J. Sched..

[27]  Tadashi Shibata,et al.  Coupled-Oscillator Associative Memory Array Operation for Pattern Recognition , 2015, IEEE Journal on Exploratory Solid-State Computational Devices and Circuits.

[28]  James A. Hutchby,et al.  Limits to binary logic switch scaling - a gedanken model , 2003, Proc. IEEE.

[29]  K. Miller.  On the Inverse of the Sum of Matrices , 1981 .

[30]  Suman Datta,et al.  Synchronization of pairwise-coupled, identical, relaxation oscillators based on metal- insulator phase transition devices: A model study , 2014 .

[31]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[32]  Suman Datta,et al.  Exploiting Synchronization Properties of Correlated Electron Devices in a Non-Boolean Computing Fabric for Template Matching , 2014, IEEE Journal on Emerging and Selected Topics in Circuits and Systems.

[33]  Gyungock Kim,et al.  Mechanism and observation of Mott transition in VO2-based two- and three-terminal devices , 2004 .

[34]  Frank C. Hoppensteadt,et al.  Pattern recognition via synchronization in phase-locked loop neural networks , 2000, IEEE Trans. Neural Networks Learn. Syst..

[35]  Zoltán Toroczkai,et al.  The Chaos Within Sudoku , 2012, Scientific Reports.

[36]  Adrian M. Ionescu,et al.  Fabrication of CMOS-compatible abrupt electronic switches based on vanadium dioxide , 2015 .

[37]  Joseph C. Culberson,et al.  Frozen development in graph coloring , 2001, Theor. Comput. Sci..

[38]  K. Roy,et al.  Boolean and non-Boolean computation with spin devices , 2012, 2012 International Electron Devices Meeting.

[39]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[40]  References , 1971 .

[41]  Sheldon B. Akers,et al.  Fault Diagnosis as a Graph Coloring Problem , 1974, IEEE Transactions on Computers.

[42]  Ralph K. Cavin,et al.  The quest for the next information processing technology , 2008 .

[43]  Frank Thomson Leighton,et al.  A Graph Coloring Algorithm for Large Scheduling Problems. , 1979, Journal of research of the National Bureau of Standards.

[44]  Hyun-Tak Kim,et al.  Observation of Mott Transition in VO_2 Based Transistors , 2003 .

[45]  B. Dickinson,et al.  The complexity of analog computation , 1986 .

[46]  Suman Datta,et al.  Intrinsic electronic switching time in ultrathin epitaxial vanadium dioxide thin film , 2013 .

[47]  Peter C. Cheeseman,et al.  Where the Really Hard Problems Are , 1991, IJCAI.

[48]  J. A. Weldon,et al.  Low-power, high-performance S-NDR oscillators for stereo (3D) vision using directly-coupled oscillator networks , 2016, 2016 IEEE Symposium on VLSI Technology.

[49]  H. Ehrke,et al.  Coherent structural dynamics and electronic correlations during an ultrafast insulator-to-metal phase transition in VO2. , 2007, Physical review letters.

[50]  S. Datta,et al.  Pairwise coupled hybrid vanadium dioxide-MOSFET (HVFET) oscillators for non-boolean associative computing , 2014, 2014 IEEE International Electron Devices Meeting.

[51]  Rui Li,et al.  Clustering dynamics of nonlinear oscillator network: Application to graph coloring problem , 2011 .

[52]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[53]  Susan Dumps,et al.  A model study. , 1988, Nursing standard (Royal College of Nursing (Great Britain) : 1987).

[54]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[55]  Chai Wah Wu,et al.  Graph coloring via synchronization of coupled oscillators , 1998 .

[56]  J. C. Kieffer,et al.  Evidence for a structurally-driven insulator-to-metal transition in VO 2 : A view from the ultrafast timescale , 2004, cond-mat/0403214.

[57]  Tad Hogg,et al.  A New Look at the Easy-Hard-Easy Pattern of Combinatorial Search Difficulty , 1997, J. Artif. Intell. Res..

[58]  Giacomo Indiveri,et al.  An event-based architecture for solving constraint satisfaction problems , 2015, Nature Communications.

[59]  Suman Datta,et al.  Nanoscale Structural Evolution of Electrically Driven Insulator to Metal Transition in Vanadium Dioxide , 2013 .

[60]  V Elser,et al.  Searching with iterated maps , 2007, Proceedings of the National Academy of Sciences.

[61]  Kaushik Roy,et al.  Magnetic Tunnel Junction Based Long-Term Short-Term Stochastic Synapse for a Spiking Neural Network with On-Chip STDP Learning , 2016, Scientific Reports.