Efficiently approximating vertex cover on scale-free networks with underlying hyperbolic geometry

The computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results: 1) A deterministic O(MIS(n,Δ)/e)-round algorithm that finds a (1+e)Δ-approximation for MaxIS in the CONGEST model. 2) A randomized (poly(log log n)/e)-round algorithm that finds, with high probability, a (1+e)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result. 3) A randomized O(log n⋅ poly(log log n)/e)-round algorithm that finds, with high probability, a 8(1+e)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+e)), this result improves upon the previous best known result in both the approximation factor and the running time. One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/e)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+e)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

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. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 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 properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N 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 or almost surely for those random graph classes. In the second part we study three classical textsf {NP}-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta )-approximation for Minimum Vertex Cover and Minimum Dominating Set, and a Omega (Delta )-approximation for Maximum Independent Set. We prove that the PLB-U property with beta >2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds.

Parameterized approximation algorithms for weighted vertex cover

We study an adjacency-list-only evolutionary algorithm (EA) for Vertex Cover on Random Hyperbolic Graphs (HRG) and Geometric Inhomogeneous Random Graphs (GIRG) with exponent τ ∈ (2, 3). The EA uses an exact uncovered-edge sampler (edges selected with

Measure & Conquer (M&C) is a prominent technique for analyzing exact algorithms for computationally hard problems, in particular, graph problems. It tries to balance worse and better situations within the algorithm analysis. This has led, e.g., to algorithms for Minimum Vertex Cover with a running time of $\mathcal{O}(c^{n})$ for some constant c≈1.2, where n is the number of vertices in the graph. Several obstacles prevent the application of this technique in parameterized algorithmics, making it rarely applied in this area. However, these difficulties can be handled in some situations. We will exemplify this with two problems related to Vertex Cover, namely Connected Vertex Cover and Edge Dominating Set. For both problems, several parameterized algorithms have been published, all based on the idea of first enumerating minimal vertex covers. Using M&C in this context will allow us to improve on the hitherto published running times. In contrast to some of the earlier suggested algorithms, ours will use polynomial space.

A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) approximately k(-gamma), are shown to exhibit qualitatively different dynamic behavior for gamma < 5/2 and gamma > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < gamma < 5/2. For 2 < gamma < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For gamma > 5/2, on the other hand, this decay time diverges as ln(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for gamma > 5/2 but to grow as N(alpha) with alpha = (5 - 2gamma)/(gamma - 1) for 2 < gamma < 5/2.

In network security, how to use efficient response methods against cascading failures of complex networks is very important. In this paper, concerned with the highest-load attack (HL) and random attack (RA) on one edge, we define five kinds of weighting strategies to assign the external resources for recovering the edges from cascading failures in heterogeneous scale-free (SF) networks. The influence of external resources, the tolerance parameter, and the different weighting strategies on SF networks against cascading failures is investigated carefully. We find that, under HL attack, the fourth kind of weighting method can more effectively improve the integral robustness of SF networks, simultaneously control the spreading velocity, and control the outburst of cascading failures in SF networks than other methods. Moreover, the third method is optimal if we only knew the local structure of SF networks and the uniform assignment is the worst. The simulations of the real-world autonomous system in, Internet have also supported our findings. The results are useful for using efficient response strategy against the emergent accidents and controlling the cascading failures in the real-world networks.

Hyperbolic random graphs (HRGs) and geometric inhomogeneous random graphs (GIRGs) are two similar generative network models that were designed to resemble complex real-world networks. In particular, they have a power-law degree distribution with controllable exponent $\beta$ and high clustering that can be controlled via the temperature $T$ .We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to $T = 0$ . We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, that is, they involve no approximation.Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input.Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straightforward inclusion does not hold in practice. However, the difference is negligible for most use cases.

Scale-free networks, as one major class of complex networks, are characterized by a specific structural feature of power-law degree distributions. Examples of scale-free real-world networks include World Wide Web (WWW), email networks, citation and co-author scientific networks, social networks (Facebook, Twitter), networks of Internet routers, and protein-protein interaction networks. In this paper we model and evaluate the reliability of scale-free complex networks using a binary decision diagram (BDD)-based method. Two-terminal reliability is considered, which is defined as the probability that a specified pair of nodes can communication through at least one fault-tree path in the network. This reliability metric can measure, for example, how reliable the information can be shared between two parties or in general how reliable the information can be disseminated within a social network. The Barabasi-Albert model is used for generating the sample scale-free networks studied in this paper. The reliability performance of scale-free networks is compared to that of random networks under the Erdos-Renyi model. Degree distributions and clustering coefficients (average fraction of pairs of neighbors of a node which are also neighbors of each other) of sample scale-free and random networks are illustrated and compared. Their effects on the performance of the BDD-based method for the reliability analysis of both scale-free and random networks are also discussed.

In the vertex cover problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of vertices S of size at most k such that every edge of G is incident on at least one vertex in S. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $$r = k-n/3$$ . We give an algorithm for this problem whose running time is $$O^*(1.6253^r)$$ . As a corollary, we obtain an $$O^*(1.2403^k)$$ -time algorithm for vertex cover on graphs with maximum degree 4.

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw). We prove that for any graph, the number of minimal separators is \(\mathcal{O}^*(3^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.6181^{\operatorname{mw}})\), the number of potential maximal cliques is \(\mathcal{O}^*(4^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\), and these objects can be listed within the same running times. (The \(\mathcal{O}^*\) notation suppresses polynomial factors in the size of the input.) Combined with known results [3,12], we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time \(\mathcal{O}^*(4^{\operatorname{vc}})\) or \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\).

We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2O(√k)nO(1). Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, clique-transversal set, and many others restricted to bounded-genus graphs. Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes as special cases all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, single-crossing-minor-free graphs, and map graphs; we extend these results to apply to bounded-genus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a bounded-genus graph that excludes some planar graph H as a minor. This bound depends linearly on the size |V (H)| of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graphminors work of Robertson & Seymour.Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. In particular, this general category of graphs includes planar graphs, bounded-genus graphs, single-crossing-minor-free graphs, and any class of graphs that is closed under taking minors. Specifically, the running time is 2O(√k)nh, where h is a constant depending only on H, which is polynomial for k = O(log2n). We introduce a general approach for developing algorithms on H-minor-free graphs, based on structural results about H-minor-free graphs at the heart of Robertson & Seymour's graph-minors work. We believe this approach opens the way to further development for problems on H-minor-free graphs.

To introduce our question and the parameterization, consider the classical Vertex Cover problem. In this problem, the input is a graph G on n vertices and a positive integer \(\ell \), and the goal is to find a vertex subset S of size at most \(\ell \) such that \(G-S\) is an independent set. Further, we want that G[S] is highly connected. That is, G[S] should be \(n-k\) edge-connected. Clearly, the problem is NP-complete, as substituting \(k=n-1\), we obtain the Connected Vertex Cover problem. A simple observation also shows that the problem admits an algorithm with running time \(n^{{\mathcal O}(k)}\). Since the problem is polynomial-time solvable for every fixed integer k, a natural parameter is the integer k. In all the problems we consider, the parameter is k, and the goal is to find a solution S of size at most \(\ell \), such that G[S] is \(n-k\) edge-connected and \(G-S\) satisfies a property. We show that this version of well-known problems such as Vertex Cover, Feedback Vertex Set, Odd Cycle Transversal and Multiway Cut admit an algorithm with running time \(f(k)\cdot n^{{\mathcal O}(1)}\), that is, they are FPT with the parameter k. One of our main subroutines to obtain these algorithms is an FPT algorithm for \(n-k\) edge connected Steiner Subgraph, which could be of an independent interest. Finally, we also show that such an algorithm is not possible for Multicut.Keywordssubset problemsparameterized algorithmsconnectivity

Parameterized algorithms for finding highly connected solution