An interacting replica approach applied to the traveling salesman problem

We present a physics inspired heuristic method for solving combinatorial optimization problems. Our approach is specifically motivated by the desire to avoid trapping in metastable local minima- a common occurrence in hard problems with multiple extrema. Our method involves (i) coupling otherwise independent simulations of a system ("replicas") via geometrical distances as well as (ii) probabilistic inference applied to the solutions found by individual replicas. The {\it ensemble} of replicas evolves as to maximize the inter-replica correlation while simultaneously minimize the local intra-replica cost function (e.g., the total path length in the Traveling Salesman Problem within each replica). We demonstrate how our method improves the performance of rudimentary local optimization schemes long applied to the NP hard Traveling Salesman Problem. In particular, we apply our method to the well-known "$k$-opt" algorithm and examine two particular cases- $k=2$ and $k=3$. With the aid of geometrical coupling alone, we are able to determine for the optimum tour length on systems up to $280$ cities (an order of magnitude larger than the largest systems typically solved by the bare $k=3$ opt). The probabilistic replica-based inference approach improves $k-opt$ even further and determines the optimal solution of a problem with $318$ cities and find tours whose total length is close to that of the optimal solutions for other systems with a larger number of cities.Comment: To appear in SAI 2016 conference proceedings 12 pages,17 figure

Near optimal configurations in mean field disordered systems

We present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems. Our approach is based on a generalization of the cavity method to a system interacting with its ground state. With this technique we study the random matching problem as well as the mean field diluted spin glass. As a byproduct of this approach we calculate the de Almeida-Thouless transition line of the spin glass on a fixed connectivity random graph.Comment: 13 pages, 7 figure

Quantum Annealing and Analog Quantum Computation

We review here the recent success in quantum annealing, i.e., optimization of the cost or energy functions of complex systems utilizing quantum fluctuations. The concept is introduced in successive steps through the studies of mapping of such computationally hard problems to the classical spin glass problems. The quantum spin glass problems arise with the introduction of quantum fluctuations, and the annealing behavior of the systems as these fluctuations are reduced slowly to zero. This provides a general framework for realizing analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of Modern Physics (in press

Physics-inspired Replica Approaches to Computer Science Problems

We study machine learning class classification problems and combinatorial optimization problems using physics inspired replica approaches. In the current work, we focus on the traveling salesman problem which is one of the most famous problems in the entire field of combinatorial optimization. Our approach is specifically motivated by the desire to avoid trapping in metastable local minima-a common occurrence in hard problems with multiple extrema. Our method involves (i) coupling otherwise independent simulations of a system (“replicas”) via geometrical distances as well as (ii) probabilistic inference applied to the solutions found by individual replicas. In particular, we apply our method to the well-known “k-opt” algorithm and examine two particular cases-k = 2 and k = 3. With the aid of geometrical coupling alone, we are able to determine for the optimum tour length on systems up to 280 cities (an order of magnitude larger than the largest systems typically solved by the bare k = 3 opt). The probabilistic replica-based inference approach improves k - opt even further and determines the optimal solution of a problem with 318 cities. In this work, we also formulate a supervised machine learning algorithm for classification problems which is called “Stochastic Replica Voting Machine” (SRVM). The method is based on the representations of known data via multiple linear expansions in terms of various stochastic functions. The algorithm is developed, implemented and applied to a binary and a 3-class classification problems in material science. Here, we employ SRVM to predict candidate compounds capable of forming cubic Perovskite structure and further classify binary (AB) solids. We demonstrated that our SRVM method exceeds the well-known Support Vector Machine (SVM) in terms of accuracy when predicting the cubic Perovskite structure. The algorithm has also been tested on 8 diverse training data sets of various types and feature space dimensions from UCI machine learning repository. It has been shown to consistently match or exceed the accuracy of existing algorithms, while simultaneously avoiding many of their pitfalls

Networking - A Statistical Physics Perspective

Efficient networking has a substantial economic and societal impact in a broad range of areas including transportation systems, wired and wireless communications and a range of Internet applications. As transportation and communication networks become increasingly more complex, the ever increasing demand for congestion control, higher traffic capacity, quality of service, robustness and reduced energy consumption require new tools and methods to meet these conflicting requirements. The new methodology should serve for gaining better understanding of the properties of networking systems at the macroscopic level, as well as for the development of new principled optimization and management algorithms at the microscopic level. Methods of statistical physics seem best placed to provide new approaches as they have been developed specifically to deal with non-linear large scale systems. This paper aims at presenting an overview of tools and methods that have been developed within the statistical physics community and that can be readily applied to address the emerging problems in networking. These include diffusion processes, methods from disordered systems and polymer physics, probabilistic inference, which have direct relevance to network routing, file and frequency distribution, the exploration of network structures and vulnerability, and various other practical networking applications.Comment: (Review article) 71 pages, 14 figure