Fast Distributed Approximation for TAP and 2-Edge-Connectivity

The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph $G$ and a spanning tree $T$ for it, and the goal is to augment $T$ with a minimum set of edges $Aug$ from $G$, such that $T \cup Aug$ is 2-edge-connected. TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and J\'{a}J\'{a}, SICOMP 1981. Recently, a 3/2-approximation was given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018], and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017; Fiorini et al., SODA 2018]. In this paper, we provide the first fast distributed approximations for TAP. We present a distributed $2$-approximation for weighted TAP which completes in $O(h)$ rounds, where $h$ is the height of $T$. When $h$ is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in $O(D+\sqrt{n}\log^*{n})$ rounds, where $n$ is the number of vertices and $D$ is the diameter of $G$. Immediate consequences of our results are an $O(D)$-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an $O(h_{MST}+\sqrt{n}\log^{*}{n})$-round 3-approximation algorithm for the weighted case, where $h_{MST}$ is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2. Finally, we complement our study with proving lower bounds for distributed approximations of TAP

Small Cuts and Connectivity Certificates: A Fault Tolerant Approach

We revisit classical connectivity problems in the {CONGEST} model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with O~(1) rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of Omega(Diam+sqrt{n}) rounds). Distributed Minimum Cut: Nanongkai and Su presented a randomized algorithm for computing a (1+epsilon)-approximation of the minimum cut using O~(D +sqrt{n}) rounds where D is the diameter of the graph. For a sufficiently large minimum cut lambda=Omega(sqrt{n}), this is tight due to Das Sarma et al. [FOCS \u2711], Ghaffari and Kuhn [DISC \u2713]. - Small Cuts: A special setting that remains open is where the graph connectivity lambda is small (i.e., constant). The only lower bound for this case is Omega(D), with a matching bound known only for lambda <= 2 due to Pritchard and Thurimella [TALG \u2711]. Recently, Daga, Henzinger, Nanongkai and Saranurak [STOC \u2719] raised the open problem of computing the minimum cut in poly(D) rounds for any lambda=O(1). In this paper, we resolve this problem by presenting a surprisingly simple algorithm, that takes a completely different approach than the existing algorithms. Our algorithm has also the benefit that it computes all minimum cuts in the graph, and naturally extends to vertex cuts as well. At the heart of the algorithm is a graph sampling approach usually used in the context of fault tolerant (FT) design. - Deterministic Algorithms: While the existing distributed minimum cut algorithms are randomized, our algorithm can be made deterministic within the same round complexity. To obtain this, we introduce a novel definition of universal sets along with their efficient computation. This allows us to derandomize the FT graph sampling technique, which might be of independent interest. - Computation of all Edge Connectivities: We also consider the more general task of computing the edge connectivity of all the edges in the graph. In the output format, it is required that the endpoints u,v of every edge (u,v) learn the cardinality of the u-v cut in the graph. We provide the first sublinear algorithm for this problem for the case of constant connectivity values. Specifically, by using the recent notion of low-congestion cycle cover, combined with the sampling technique, we compute all edge connectivities in poly(D) * 2^{O(sqrt{log n log log n})} rounds. Sparse Certificates: For an n-vertex graph G and an integer lambda, a lambda-sparse certificate H is a subgraph H subseteq G with O(lambda n) edges which is lambda-connected iff G is lambda-connected. For D-diameter graphs, constructions of sparse certificates for lambda in {2,3} have been provided by Thurimella [J. Alg. \u2797] and Dori [PODC \u2718] respectively using O~(D) number of rounds. The problem of devising such certificates with o(D+sqrt{n}) rounds was left open by Dori [PODC \u2718] for any lambda >= 4. Using connections to fault tolerant spanners, we considerably improve the round complexity for any lambda in [1,n] and epsilon in (0,1), by showing a construction of (1-epsilon)lambda-sparse certificates with O(lambda n) edges using only O(1/epsilon^2 * log^{2+o(1)} n) rounds

Exact mean field inference in asymmetric kinetic Ising systems

We develop an elementary mean field approach for fully asymmetric kinetic Ising models, which can be applied to a single instance of the problem. In the case of the asymmetric SK model this method gives the exact values of the local magnetizations and the exact relation between equal-time and time-delayed correlations. It can also be used to solve efficiently the inverse problem, i.e. determine the couplings and local fields from a set of patterns, also in cases where the fields and couplings are time-dependent. This approach generalizes some recent attempts to solve this dynamical inference problem, which were valid in the limit of weak coupling. It provides the exact solution to the problem also in strongly coupled problems. This mean field inference can also be used as an efficient approximate method to infer the couplings and fields in problems which are not infinite range, for instance in diluted asymmetric spin glasses.Comment: 10 pages, 7 figure

An Empirical Analysis of Approximation Algorithms for the Unweighted Tree Augmentation Problem

In this thesis, we perform an experimental study of approximation algorithms for the tree augmentation problem (TAP). TAP is a fundamental problem in network design. The goal of TAP is to add the minimum number of edges from a given edge set to a tree so that it becomes 2-edge connected. Formally, given a tree T = (V, E), where V denotes the set of vertices and E denotes the set of edges in the tree, and a set of edges (or links) L ⊆ V × V disjoint from E, the objective is to find a set of edges to add to the tree F ⊆ L such that the augmented tree (V, E ∪ F) is 2-edge connected. Our goal is to establish a baseline performance for each approximation algorithm on actual instances rather than worst-case instances. In particular, we are interested in whether the algorithms rank on practical instances is consistent with their worst-case guarantee rankings. We are also interested in whether preprocessing times, implementation difficulties, and running times justify the use of an algorithm in practice. We profiled and analyzed five approximation algorithms, viz., the Frederickson algorithm, the Nagamochi algorithm, the Even algorithm, the Adjiashivili algorithm, and the Grandoni algorithm. Additionally, we used an integer program and a simple randomized algorithm as benchmarks. The performance of each algorithm was measured using space, time, and quality comparison metrics. We found that the simple randomized is competitive with the approximation algorithms and that the algorithms rank according to their theoretical guarantees. The randomized algorithm is simpler to implement and understand. Furthermore, the randomized algorithm runs faster and uses less space than any of the more sophisticated approximation algorithms

Evolving graphs: dynamical models, inverse problems and propagation

Applications such as neuroscience, telecommunication, online social networking, transport and retail trading give rise to connectivity patterns that change over time. In this work, we address the resulting need for network models and computational algorithms that deal with dynamic links. We introduce a new class of evolving range-dependent random graphs that gives a tractable framework for modelling and simulation. We develop a spectral algorithm for calibrating a set of edge ranges from a sequence of network snapshots and give a proof of principle illustration on some neuroscience data. We also show how the model can be used computationally and analytically to investigate the scenario where an evolutionary process, such as an epidemic, takes place on an evolving network. This allows us to study the cumulative effect of two distinct types of dynamics