Algorithmic Complexity of Power Law Networks

It was experimentally observed that the majority of real-world networks follow power law degree distribution. The aim of this paper is to study the algorithmic complexity of such "typical" networks. The contribution of this work is twofold. First, we define a deterministic condition for checking whether a graph has a power law degree distribution and experimentally validate it on real-world networks. This definition allows us to derive interesting properties of power law networks. We observe that for exponents of the degree distribution in the range $[1,2]$ such networks exhibit double power law phenomenon that was observed for several real-world networks. Our observation indicates that this phenomenon could be explained by just pure graph theoretical properties. The second aim of our work is to give a novel theoretical explanation why many algorithms run faster on real-world data than what is predicted by algorithmic worst-case analysis. We show how to exploit the power law degree distribution to design faster algorithms for a number of classical P-time problems including transitive closure, maximum matching, determinant, PageRank and matrix inverse. Moreover, we deal with the problems of counting triangles and finding maximum clique. Previously, it has been only shown that these problems can be solved very efficiently on power law graphs when these graphs are random, e.g., drawn at random from some distribution. However, it is unclear how to relate such a theoretical analysis to real-world graphs, which are fixed. Instead of that, we show that the randomness assumption can be replaced with a simple condition on the degrees of adjacent vertices, which can be used to obtain similar results. As a result, in some range of power law exponents, we are able to solve the maximum clique problem in polynomial time, although in general power law networks the problem is NP-complete

Greed is Good for Deterministic Scale-Free Networks

Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. In fact, the behavior of real-world networks and random graph models can be the complete opposite of one another, depending on the considered property. Brach, Cygan, Lacki, and Sankowski [SODA 2016] introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both deterministic properties and exploit them to design faster algorithms for a number of classical graph problems like transitive closure, maximum matching, determinant, PageRank, matrix inverse, counting triangles and maximum clique. We complement the work of Brach et al. by showing that some well-studied random graph models exhibit both the mentioned PLB properties and additionally also a power-law lower bound on the degree distribution (PLB-L). All three properties hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. In the second part of this work we study three classical NP-hard combinatorial optimization problems on PLB networks. It is known that on general graphs, a greedy algorithm, which chooses nodes in the order of their degree, only achieves an approximation factor of asymptotically at least logarithmic in the maximum degree for Minimum Vertex Cover and Minimum Dominating Set, and an approximation factor of asymptotically at least the maximum degree for Maximum Independent Set. We prove that the PLB-U property suffices such that the greedy approach achieves a constant-factor approximation for all three problems. We also show that all three combinatorial optimization problems are APX-complete, even if all PLB-properties hold. Hence, a PTAS cannot be expected, unless P=NP

Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations

Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem'' when applied to fermions - causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.Comment: 4 page