Solving the undirected feedback vertex set problem by local search

An undirected graph consists of a set of vertices and a set of undirected edges between vertices. Such a graph may contain an abundant number of cycles, then a feedback vertex set (FVS) is a set of vertices intersecting with each of these cycles. Constructing a FVS of cardinality approaching the global minimum value is a optimization problem in the nondeterministic polynomial-complete complexity class, therefore it might be extremely difficult for some large graph instances. In this paper we develop a simulated annealing local search algorithm for the undirected FVS problem. By defining an order for the vertices outside the FVS, we replace the global cycle constraints by a set of local vertex constraints on this order. Under these local constraints the cardinality of the focal FVS is then gradually reduced by the simulated annealing dynamical process. We test this heuristic algorithm on large instances of Er\"odos-Renyi random graph and regular random graph, and find that this algorithm is comparable in performance to the belief propagation-guided decimation algorithm.Comment: 6 page

Nearly deconfined spinon excitations in the square-lattice spin-1/2 Heisenberg antiferromagnet

We study the spin-excitation spectrum (dynamic structure factor) of the spin-1/2 square-lattice Heisenberg antiferromagnet and an extended model (the J−Q model) including four-spin interactions Q in addition to the Heisenberg exchange J. Using an improved method for stochastic analytic continuation of imaginary-time correlation functions computed with quantum Monte Carlo simulations, we can treat the sharp (δ-function) contribution to the structure factor expected from spin-wave (magnon) excitations, in addition to resolving a continuum above the magnon energy. Spectra for the Heisenberg model are in excellent agreement with recent neutron-scattering experiments on Cu(DCOO)2⋅4D2O, where a broad spectral-weight continuum at wave vector q=(π,0) was interpreted as deconfined spinons, i.e., fractional excitations carrying half of the spin of a magnon. Our results at (π,0) show a similar reduction of the magnon weight and a large continuum, while the continuum is much smaller at q=(π/2,π/2) (as also seen experimentally). We further investigate the reasons for the small magnon weight at (π,0) and the nature of the corresponding excitation by studying the evolution of the spectral functions in the J−Q model. Upon turning on the Q interaction, we observe a rapid reduction of the magnon weight to zero, well before the system undergoes a deconfined quantum phase transition into a nonmagnetic spontaneously dimerized state. Based on these results, we reinterpret the picture of deconfined spinons at (π,0) in the experiments as nearly deconfined spinons—a precursor to deconfined quantum criticality. To further elucidate the picture of a fragile (π,0)-magnon pole in the Heisenberg model and its depletion in the J−Q model, we introduce an effective model of the excitations in which a magnon can split into two spinons that do not separate but fluctuate in and out of the magnon space (in analogy to the resonance between a photon and a particle-hole pair in the exciton-polariton problem). The model can reproduce the reduction of magnon weight and lowered excitation energy at (π,0) in the Heisenberg model, as well as the energy maximum and smaller continuum at (π/2,π/2). It can also account for the rapid loss of the (π,0) magnon with increasing Q and the remarkable persistence of a large magnon pole at q=(π/2,π/2) even at the deconfined critical point. The fragility of the magnons close to (π,0) in the Heisenberg model suggests that various interactions that likely are important in many materials—e.g., longer-range pair exchange, ring exchange, and spin-phonon interactions—may also destroy these magnons and lead to even stronger spinon signatures than in Cu(DCOO)2⋅4D2O.We thank Wenan Guo, Akiko Masaki-Kato, Andrey Mishchenko, Martin Mourigal, Henrik Ronnow, Kai Schmidt, Cenke Xu, and Seiji Yunoki for useful discussions. Experimental data from Ref. [33] were kindly provided by N. B. Christensen and H. M. Ronnow. H. S. was supported by the China Postdoctoral Science Foundation under Grants No. 2016M600034 and No. 2017T100031. St.C. was funded by the NSFC under Grants No. 11574025 and No. U1530401. Y. Q. Q. and Z. Y. M. acknowledge funding from the Ministry of Science and Technology of China through National Key Research and Development Program under Grant No. 2016YFA0300502, from the key research program of the Chinese Academy of Sciences under Grant No. XDPB0803, and from the NSFC under Grants No. 11421092, No. 11574359, and No. 11674370, as well as the National Thousand-Young Talents Program of China. A. W. S. was funded by the NSF under Grants No. DMR-1410126 and No. DMR-1710170, and by the Simons Foundation. In addition H. S., Y. Q. Q., and Sy. C. thank Boston University's Condensed Matter Theory Visitors program for support, and A. W. S. thanks the Beijing Computational Science Research Center and the Institute of Physics, Chinese Academy of Sciences for visitor support. We thank the Center for Quantum Simulation Sciences at the Institute of Physics, Chinese Academy of Sciences, the Tianhe-1A platform at the National Supercomputer Center in Tianjin, Boston University's Shared Computing Cluster, and CALMIP (Toulouse) for their technical support and generous allocation of CPU time. (2016M600034 - China Postdoctoral Science Foundation; 2017T100031 - China Postdoctoral Science Foundation; 11574025 - NSFC; U1530401 - NSFC; 11421092 - NSFC; 11574359 - NSFC; 11674370 - NSFC; 2016YFA0300502 - Ministry of Science and Technology of China; XDPB0803 - Chinese Academy of Sciences; National Thousand-Young Talents Program of China; DMR-1410126 - NSF; DMR-1710170 - NSF; Simons Foundation; Boston University's Condensed Matter Theory Visitors program)Accepted manuscript and published version

Patterns, Entropy, and Predictability of Human Mobility and Life

Cellular phones are now offering an ubiquitous means for scientists to observe life: how people act, move and respond to external influences. They can be utilized as measurement devices of individual persons and for groups of people of the social context and the related interactions. The picture of human life that emerges shows complexity, which is manifested in such data in properties of the spatiotemporal tracks of individuals. We extract from smartphone-based data for a set of persons important locations such as “home”, “work” and so forth over fixed length time-slots covering the days in the data-set (see also [1], [2]). This set of typical places is heavy-tailed, a power-law distribution with an exponent close to −1.7. To analyze the regularities and stochastic features present, the days are classified for each person into regular, personal patterns. To this are superimposed fluctuations for each day. This randomness is measured by “life” entropy, computed both before and after finding the clustering so as to subtract the contribution of a number of patterns. The main issue that we then address is how predictable individuals are in their mobility. The patterns and entropy are reflected in the predictability of the mobility of the life both individually and on average. We explore the simple approaches to guess the location from the typical behavior, and of exploiting the transition probabilities with time from location or activity A to B. The patterns allow an enhanced predictability, at least up to a few hours into the future from the current location. Such fixed habits are most clearly visible in the working-day length.Peer reviewe