Typical Performance of Approximation Algorithms for NP-hard Problems

Typical performance of approximation algorithms is studied for randomized minimum vertex cover problems. A wide class of random graph ensembles characterized by an arbitrary degree distribution is discussed with some theoretical frameworks. Here three approximation algorithms are examined; the linear-programming relaxation, the loopy-belief propagation, and the leaf-removal algorithm. The former two algorithms are analyzed using the statistical-mechanical technique while the average-case analysis of the last one is studied by the generating function method. These algorithms have a threshold in the typical performance with increasing the average degree of the random graph, below which they find true optimal solutions with high probability. Our study reveals that there exist only three cases determined by the order of the typical-performance thresholds. We provide some conditions for classifying the graph ensembles and demonstrate explicitly examples for the difference in the threshold.Comment: 21 pages, 5 figures; typos are fixe

Fault Tolerance of Random Graphs with respect to Connectivity: Mean-field Approximation for Semi-dense Random Graphs

The fault tolerance of random graphs with unbounded degrees with respect to connectivity is investigated, which relates to the reliability of wireless sensor networks with unreliable relay nodes. The model evaluates the network breakdown probability that a graph is disconnected after stochastic node removal. To establish a mean-field approximation for the model, we propose the cavity method for finite systems. The analysis enables us to obtain an approximation formula for random graphs with any number of nodes and an arbitrary degree distribution. In addition, its asymptotic analysis reveals that the phase transition occurs in semi-dense random graphs whose average degree grows logarithmically. These results, which are supported by numerical simulations, coincide with the mathematical results, indicating successful predictions by mean-field approximation for unbounded but not dense random graphs.Comment: 5 pages, 3 figure

Typical Approximation Performance for Maximum Coverage Problem

This study investigated typical performance of approximation algorithms known as belief propagation, greedy algorithm, and linear-programming relaxation for maximum coverage problems on sparse biregular random graphs. After using the cavity method for a corresponding hard-core lattice--gas model, results show that two distinct thresholds of replica-symmetry and its breaking exist in the typical performance threshold of belief propagation. In the low-density region, the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses. Although the greedy algorithm and linear-programming relaxation have the same approximation ratio in worst-case performance, their typical performance thresholds are mutually different, indicating the importance of typical performance. Results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms.Comment: 10 pages, 6 figure

Non-relativistic Collisionless Shocks in Unmagnetized Electron-Ion Plasmas

We show that the Weibel-mediated collisionless shocks are driven at non-relativistic propagation speed (0.1c < V < 0.45c) in unmagnetized electron-ion plasmas by performing two-dimensional particle-in-cell simulations. It is shown that the profiles of the number density and the mean velocity in the vicinity of the shock transition region, which are normalized by the respective upstream values, are almost independent of the upstream bulk velocity, i.e., the shock velocity. In particular, the width of the shock transition region is ~100 ion inertial length independent of the shock velocity. For these shocks the energy density of the magnetic field generated by the Weibel-type instability within the shock transition region reaches typically 1-2% of the upstream bulk kinetic energy density. This mechanism probably explains the robust formation of collisionless shocks, for example, driven by young supernova remnants, with no assumption of external magnetic field in the universe.Comment: 4 pages, 7 figures, accepted for publication in ApJ Letter