Graph optimisation problems and the Potts glass

Fluctuations in the random-link matching problem

Using the replica approach and the cavity method, we study the fluctuations of the optimal cost in the random-link matching problem. By means of replica arguments, we derive the exact expression of its variance. Moreover, we study the large deviation function, deriving its expression in two different ways, namely using both the replica method and the cavity method.

Statistical physics of hard optimization problems

Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the non-deterministic polynomial (NP)-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this article is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.

(Un)detectable cluster structure in sparse networks

Can a cluster structure in a sparse relational data set, i.e., a network, be detected at all by unsupervised clustering techniques? We answer this question by means of statistical mechanics making our analysis independent of any particular algorithm used for clustering. We find a sharp transition from a phase in which the cluster structure is not detectable at all to a phase in which it can be detected with high accuracy. We calculate the transition point and the shape of the transition, i.e., the theoretically achievable accuracy, analytically. This illuminates theoretical limitations of data mining in networks and allows for an understanding and evaluation of the performance of a variety of algorithms.

Partitioning and modularity of graphs with arbitrary degree distribution

We solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut size to scale universally with <square root k> . In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with square root <k> [Fu and Anderson, J. Phys. A 19, 1605 (1986)]. Our results also generalize to the problem of q partitioning. They can be used to find the expected modularity Q [Newman and Girvan, Phys. Rev. E 69, 026113 (2004)] of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms.

Statistical mechanics of community detection

Starting from a general ansatz, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the ad hoc introduced quality function from [J. Reichardt and S. Bornholdt, Phys. Rev. Lett. 93, 218701 (2004)] and the modularity Q as defined by Newman and Girvan [Phys. Rev. E 69, 026113 (2004)] as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further, we show how hierarchies and overlap in the community structure can be detected. Computationally efficient local update rules for optimization procedures to find the ground state are given. We show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure.

Polynomial iterative algorithms for coloring and analyzing random graphs

We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on q, we find with a one-step replica-symmetry breaking approximation the precise value of the critical average connectivity c(q). Moreover, we show that below c(q) there exists a clustering phase c in [c(d),c(q)] in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region, i.e., when c in [c(d),c(q)].

Phase transition in multiprocessor scheduling

An "easy-hard" phase transition is shown to characterize the multiprocessor scheduling problem in which one has to distribute the workload on a parallel computer such as to minimize the overall run time. The transition can be analyzed in detail by mapping it on a mean-field antiferromagnetic Potts model. The static phase transition, characterized by a vanishing ground state entropy, corresponds to a transition in the performance of practical scheduling algorithms.

Coloring random graphs

We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring, whereas graphs with high connectivity are uncolorable. Depending on q, we find the precise value of the critical average connectivity c(q). Moreover, we show that below c(q) there exists a clustering phase c in [c(d),c(q)] in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.

Parallel Distributed Approaches to Combinatorial Optimization: Benchmark Studies on Traveling Salesman Problem

We present and summarize the results from 50-, 100-, and 200-city TSP benchmarks presented at the 1989 Neural Information Processing Systems (NIPS) postconference workshop using neural network, elastic net, genetic algorithm, and simulated annealing approaches. These results are also compared with a state-of-the-art hybrid approach consisting of greedy solutions, exhaustive search, and simulated annealing.