arXiv subjects

Ruben Becker

Publications and source records attributed to Ruben Becker.

28 records · Page 2Linked to original sources

arXiv2020

Group-Harmonic and Group-Closeness Maximization -- Approximation and Engineering

Centrality measures characterize important nodes in networks. Efficiently computing such nodes has received a lot of attention. When considering the generalization of computing central groups of nodes, challenging optimization problems occur. In this work, we study two such problems, group-harmonic maximization and group-closeness maximization both from a theoretical and from an algorithm engineering perspective. On the theoretical side, we obtain the following results. For group-harmonic maximization, unless $P=NP$, there is no polynomial-time algorithm that achieves an approximation factor better than $1-1/e$ (directed) and $1-1/(4e)$ (undirected), even for unweighted graphs. On the positive side, we show that a greedy algorithm achieves an approximation factor of $λ(1-2/e)$ (directed) and $λ(1-1/e)/2$ (undirected), where $λ$ is the ratio of minimal and maximal edge weights. For group-closeness maximization, the undirected case is $NP$-hard to be approximated to within a factor better than $1-1/(e+1)$ and a constant approximation factor is achieved by a local-search algorithm. For the directed case, however, we show that, for any $ε<1/2$, the problem is $NP$-hard to be approximated within a factor of $4|V|^{-ε}$. From the algorithm engineering perspective, we provide efficient implementations of the above greedy and local search algorithms. In our experimental study we show that, on small instances where an optimum solution can be computed in reasonable time, the quality of both the greedy and the local search algorithms come very close to the optimum. On larger instances, our local search algorithms yield results with superior quality compared to existing greedy and local search solutions, at the cost of additional running time. We thus advocate local search for scenarios where solution quality is of highest concern.

cs.DS↗

arXiv2020

Maximizing Influence-based Group Shapley Centrality

One key problem in network analysis is the so-called influence maximization problem, which consists in finding a set $S$ of at most $k$ seed users, in a social network, maximizing the spread of information from $S$. This paper studies a related but slightly different problem: We want to find a set $S$ of at most $k$ seed users that maximizes the spread of information, when $S$ is added to an already pre-existing - but unknown - set of seed users $T$. We consider such scenario to be very realistic. Assume a central entity wants to spread a piece of news, while having a budget to influence $k$ users. This central authority may know that some users are already aware of the information and are going to spread it anyhow. The identity of these users being however completely unknown. We model this optimization problem using the Group Shapley value, a well-founded concept from cooperative game theory. While the standard influence maximization problem is easy to approximate within a factor $1-1/e-ε$ for any $ε>0$, assuming common computational complexity conjectures, we obtain strong hardness of approximation results for the problem at hand in this paper. Maybe most prominently, we show that it cannot be approximated within $1/n^{o(1)}$ under the Gap Exponential Time Hypothesis. Hence, it is unlikely to achieve anything better than a polynomial factor approximation. Nevertheless, we show that a greedy algorithm can achieve a factor of $\frac{1-1/e}{k}-ε$ for any $ε>0$, showing that not all is lost in settings where $k$ is bounded.

cs.GT↗

arXiv2019

Low Diameter Graph Decompositions by Approximate Distance Computation

In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of such decompositions inherently rely on the subtractive form of the triangle inequality. The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 13), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the Congest, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 96) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only O(log n) times, which is of interest for capacitated problems and simulating Congest algorithms on the tree into which the graph is embedded.

cs.DC↗

arXiv2019

Subspace Determination through Local Intrinsic Dimensional Decomposition: Theory and Experimentation

Axis-aligned subspace clustering generally entails searching through enormous numbers of subspaces (feature combinations) and evaluation of cluster quality within each subspace. In this paper, we tackle the problem of identifying subsets of features with the most significant contribution to the formation of the local neighborhood surrounding a given data point. For each point, the recently-proposed Local Intrinsic Dimension (LID) model is used in identifying the axis directions along which features have the greatest local discriminability, or equivalently, the fewest number of components of LID that capture the local complexity of the data. In this paper, we develop an estimator of LID along axis projections, and provide preliminary evidence that this LID decomposition can indicate axis-aligned data subspaces that support the formation of clusters.

cs.LG↗

arXiv2019

Balancing spreads of influence in a social network

The personalization of our news consumption on social media has a tendency to reinforce our pre-existing beliefs instead of balancing our opinions. This finding is a concern for the health of our democracies which rely on an access to information providing diverse viewpoints. To tackle this issue from a computational perspective, Garimella et al. (NIPS'17) modeled the spread of these viewpoints, also called campaigns, using the well-known independent cascade model and studied an optimization problem that aims at balancing information exposure in a social network when two opposing campaigns propagate in the network. The objective in their $NP$-hard optimization problem is to maximize the number of people that are exposed to either both or none of the viewpoints. For two different settings, one corresponding to a model where campaigns spread in a correlated manner, and a second one, where the two campaigns spread in a heterogeneous manner, they provide constant ratio approximation algorithms. In this paper, we investigate a more general formulation of this problem. That is, we assume that $μ$ different campaigns propagate in a social network and we aim to maximize the number of people that are exposed to either $ν$ or none of the campaigns, where $μ\geν\ge2$. We provide dedicated approximation algorithms for both the correlated and heterogeneous settings. Interestingly, for the heterogeneous setting with $ν\ge 3$, we give a reduction leading to several approximation hardness results. Maybe most importantly, we obtain that the problem cannot be approximated within a factor of $n^{-g(n)}$ for any $g(n)=o(1)$ assuming Gap-ETH, denoting with $n$ the number of nodes in the social network. For $ν\ge 4$, there is no $n^{-ε}$-approximation algorithm if a certain class of one-way functions exists, where $ε> 0$ is a given constant which depends on $ν$.

cs.SI↗

arXiv2018

Two Results on Slime Mold Computations

We present two results on slime mold computations. In wet-lab experiments (Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum demonstrated its ability to solve shortest path problems. Biologists proposed a mathematical model, a system of differential equations, for the slime's adaption process (J. Theoretical Biology'07). It was shown that the process convergences to the shortest path (J. Theoretical Biology'12) for all graphs. We show that the dynamics actually converges for a much wider class of problems, namely undirected linear programs with a non-negative cost vector. Combinatorial optimization researchers took the dynamics describing slime behavior as an inspiration for an optimization method and showed that its discretization can $\varepsilon$-approximately solve linear programs with positive cost vector (ITCS'16). Their analysis requires a feasible starting point, a step size depending linearly on $\varepsilon$, and a number of steps with quartic dependence on $\mathrm{opt}/(\varepsilonΦ)$, where $Φ$ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost ($\mathrm{opt}$). We give a refined analysis showing that the dynamics initialized with any strongly dominating point converges to the set of optimal solutions. Moreover, we strengthen the convergence rate bounds and prove that the step size is independent of $\varepsilon$, and the number of steps depends logarithmically on $1/\varepsilon$ and quadratically on $\mathrm{opt}/Φ$.

cs.DS↗

arXiv2017

Counting Solutions of a Polynomial System Locally and Exactly

We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system $f_1=\cdots=f_n=0$, with $f_i\in\mathbb{C}[x_1,\ldots,x_n]$, and a polydisc $\mathbfΔ\subset\mathbb{C}^n$, our method aims to certify the existence of $k$ solutions (counted with multiplicity) within the polydisc. In case of success, it yields the correct result under guarantee. Otherwise, no information is given. However, we show that our algorithm always succeeds if $\mathbfΔ$ is sufficiently small and well-isolating for a $k$-fold solution $\mathbf{z}$ of the system. Our analysis of the algorithm further yields a bound on the size of the polydisc for which our algorithm succeeds under guarantee. This bound depends on local parameters such as the size and multiplicity of $\mathbf{z}$ as well as the distances between $\mathbf{z}$ and all other solutions. Efficiency of our method stems from the fact that we reduce the problem of counting the roots in $\mathbfΔ$ of the original system to the problem of solving a truncated system of degree $k$. In particular, if the multiplicity $k$ of $\mathbf{z}$ is small compared to the total degrees of the polynomials $f_i$, our method considerably improves upon known complete and certified methods. For the special case of a bivariate system, we report on an implementation of our algorithm, and show experimentally that our algorithm leads to a significant improvement, when integrated as inclusion predicate into an elimination method.

cs.SC↗

arXiv2016

An Integer Interior Point Method for Min-Cost Flow Using Arc Contractions and Deletions

We present an interior point method for the min-cost flow problem that uses arc contractions and deletions to steer clear from the boundary of the polytope when path-following methods come too close. We obtain a randomized algorithm running in expected $\tilde O( m^{3/2} )$ time that only visits integer lattice points in the vicinity of the central path of the polytope. This enables us to use integer arithmetic like classical combinatorial algorithms typically do. We provide explicit bounds on the size of the numbers that appear during all computations. By presenting an integer arithmetic interior point algorithm we avoid the tediousness of floating point error analysis and achieve a method that is guaranteed to be free of any numerical issues. We thereby eliminate one of the drawbacks of numerical methods in contrast to combinatorial min-cost flow algorithms that still yield the most efficient implementations in practice, despite their inferior worst-case time complexity.

cs.DS↗

arXiv2016

A Near-Optimal Subdivision Algorithm for Complex Root Isolation based on the Pellet Test and Newton Iteration

We describe a subdivision algorithm for isolating the complex roots of a polynomial $F\in\mathbb{C}[x]$. Given an oracle that provides approximations of each of the coefficients of $F$ to any absolute error bound and given an arbitrary square $\mathcal{B}$ in the complex plane containing only simple roots of $F$, our algorithm returns disjoint isolating disks for the roots of $F$ in $\mathcal{B}$. Our complexity analysis bounds the absolute error to which the coefficients of $F$ have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square $\mathcal{B}$, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the \emph{benchmark problem}, namely, to isolate all the roots of a polynomial of degree $n$ with integer coefficients of bit size less than $τ$, our algorithm needs $\tilde O(n^3+n^2τ)$ bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schönhage's splitting circle technique. Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a "soft-test" to count the number of roots in a disk. Using Schröder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm.

math.NA↗

arXiv2014

A Combinatorial $\tilde{O}(m^{3/2})$-time Algorithm for the Min-Cost Flow Problem

We present a combinatorial method for the min-cost flow problem and prove that its expected running time is bounded by $\tilde O(m^{3/2})$. This matches the best known bounds, which previously have only been achieved by numerical algorithms or for special cases. Our contribution contains three parts that might be interesting in their own right: (1) We provide a construction of an equivalent auxiliary network and interior primal and dual points with potential $P_0=\tilde{O}(\sqrt{m})$ in linear time. (2) We present a combinatorial potential reduction algorithm that transforms initial solutions of potential $P_0$ to ones with duality gap below $1$ in $\tilde O(P_0\cdot \mbox{CEF}(n,m,ε))$ time, where $ε^{-1}=O(m^2)$ and $\mbox{CEF}(n,m,ε)$ denotes the running time of any combinatorial algorithm that computes an $ε$-approximate electrical flow. (3) We show that solutions with duality gap less than $1$ suffice to compute optimal integral potentials in $O(m+n\log n)$ time with our novel crossover procedure. All in all, using a variant of a state-of-the-art $ε$-electrical flow solver, we obtain an algorithm for the min-cost flow problem running in $\tilde O(m^{3/2})$.

cs.DS↗