Ruben Becker

Complexity Analysis of Root Clustering for a Complex Polynomial

Let F(z) be an arbitrary complex polynomial. We introduce the {local root clustering problem}, to compute a set of natural epsilon-clusters of roots of F(z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of F(z) are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schonhages splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models.

We present a method for solving the shortest transshipment problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of (1 + epsilon) in undirected graphs with non-negative integer edge weights using a tailored gradient descent algorithm. Our gradient descent algorithm takes epsilon^(-3) polylog(n) iterations, and in each iteration it needs to solve an instance of the transshipment problem up to a multiplicative error of polylog(n), where n is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a careful white-box analysis, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve prior work by obtaining the following results: (1) Broadcast CONGEST model: (1 + epsilon)-approximate SSSP using ~O((sqrt(n) + D) epsilon^(-O(1))) rounds, where D is the (hop) diameter of the network. (2) Broadcast congested clique model: (1 + epsilon)-approximate shortest transshipment and SSSP using ~O(epsilon^(-O(1))) rounds. (3) Multipass streaming model: (1 + epsilon)-approximate shortest transshipment and SSSP using ~O(n) space and ~O(epsilon^(-O(1))) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs for traversing an edge in opposite directions, running times scale with the maximum ratio between the costs of both directions over all edges.

From DQBF to QBF by Dependency Elimination

In this paper, we propose the elimination of dependencies to convert a given dependency quantified Boolean formula (DQBF) to an equisatisfiable QBF. We show how to select a set of dependencies to eliminate such that we arrive at a smallest equisatisfiable QBF in terms of existential variables that is achievable using dependency elimination. This approach is improved by taking so-called don’t-care dependencies into account, which result from the application of dependency schemes to the formula and can be added to or removed from the formula at no cost. We have implemented this new method in the state-of-the-art DQBF solver HQS. Experiments show that dependency elimination is clearly superior to the previous method using variable elimination.

Two Results on Slime Mold Computations

Abstract We present two results on slime mold computations. In wet-lab experiments by Nakagaki et al. (2000) [1] 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 slimes adaption process (Tero et al., 2007) [3] . It was shown that the process convergences to the shortest path (Bonifaci et al., 2012) [5] 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 e-approximately solve linear programs with positive cost vector (Straszak and Vishnoi, 2016) [14] . Their analysis requires a feasible starting point, a step size depending linearly on e, and a number of steps with quartic dependence on opt / ( e Φ ) , where Φ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost (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 e, and the number of steps depends logarithmically on 1 / e and quadratically on opt / Φ .

A Simple Efficient Interior Point Method for Min-Cost Flow

We present a novel simpler 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 have previously been achieved only by far more complex algorithms or by algorithms for special cases. Our contribution contains three algorithmic parts that are interesting in their own right: (1) We provide a linear time construction of an equivalent auxiliary network and interior primal and dual points, i.e. flows, node potentials and slacks, with potential \(P_0=\tilde{O}(\sqrt{m})\). (2) We present a potential reduction algorithm that transforms initial solutions of potential \(P_0\) to ones with duality gap below \(1\) in \(\tilde{O}(P_0\cdot \text{ CEF }(n,m,\epsilon ))\) time, where \(\epsilon ^{-1}=O(m^2)\) and \(\text{ CEF }(n,m,\epsilon )\) denotes the running time of any algorithm that computes an \(\varepsilon \)-approximate electrical flow. (3) We show that, taking solutions with duality gap less than \(1\) as input, one can compute optimal integral node potentials in \(O(m+n\log n)\) time with our novel crossover procedure. Altogether, using a variant of a state-of-the-art \(\varepsilon \)-electrical flow solver, we obtain a new simple algorithm for the min-cost flow problem running in \(\tilde{O}(m^{3/2})\).