Matthias Poloczek

Randomized Greedy Algorithms for the Maximum Matching Problem with New Analysis

It is a long-standing problem to lower bound the performance of randomized greedy algorithms for maximum matching. Aronson, Dyer, Frieze and Suen [1]studied the modified randomized greedy (MRG) algorithm and proved that it approximates the maximum matching within a factor of at least 1/2 + 1/400,000. They use heavy combinatorial methods in their analysis. We introduce a new technique we call Contrast Analysis, and show a 1/2 + 1/256 performance lower bound for the MRG algorithm. The technique seems to be useful not only for the MRG, but also for other related algorithms.

Bounds on greedy algorithms for MAX SAT

We study adaptive priority algorithms for MAX SAT and show that no such deterministic algorithm can reach approximation ratio 3/4, assuming an appropriate model of data items. As a consequence we obtain that the Slack-Algorithm of [13] cannot be derandomized. Moreover, we present a significantly simpler version of the Slack-Algorithm and also simplify its analysis. Additionally, we show that the algorithm achieves a ratio of 3/4 even if we compare its score with the optimal fractional score.

Contagious Sets in Dense Graphs

We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m(G,r) be the size of a smallest contagious set in a graph G on n vertices. We examine density conditions that ensure m(G,r) = r for all r >= 2. With respect to the minimum degree, we prove that such a smallest possible contagious set is guaranteed to exist if and only if G has minimum degree at least (k-1)/k * n. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such a graph. We also investigate what average degree asserts the existence of small contagious sets. For n >= k >= r, we denote by M(n,k,r) the maximum number of edges in an n-vertex graph G satisfying m(G,r)>k. We determine the precise value of M(n,k,2) and M(n,k,k), assuming that n is sufficiently large compared to k.

Greedy Algorithms for the Maximum Satisfiability Problem: Simple Algorithms and Inapproximability Bounds

We give a simple, randomized greedy algorithm for the maximum satisfiability problem (MAX SAT) that obtains a

On Some Recent Approximation Algorithms for MAX SAT

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Warm starting Bayesian optimization

-approximation in expectation. In contrast to previously known

Contagious sets in dense graphs

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On Some Recent MAX SAT Approximation Algorithms

-approximation algorithms, our algorithm does not use flows or linear programming. Hence we provide a positive answer to a question posed by Williamson in 1998 on whether such an algorithm exists. Moreover, we show that Johnsons greedy algorithm cannot guarantee a

Efficient search of compositional space for hybrid organic–inorganic perovskites via Bayesian optimization

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Advances in Bayesian Optimization with Applications in Aerospace Engineering

-approximation, even if the variables are processed in a random order. Thereby we partially solve a problem posed by Chen, Friesen, and Zheng in 1999. In order to explore the limitations of the greedy paradigm, we use the model of priority algorithms of Borodin, Nielsen, and Rackoff. Since our greedy algorithm works in an online scenario where the variables arrive with their set of undecided clauses, we wonder if a better approximation ratio can be obtained by further fine-tuning its random decisions. For a particular information model ...