We consider two-player zero-sum stochastic mean payoff games with perfect information. We show that any such game, with a constant number of random positions and polynomially bounded positive transition probabilities, admits a polynomial time approximation scheme, both in the relative and absolute sense.

This thesis deals with two rumor spreading problems. In the first part, we study the rumor spreading problem in social networks modelled by preferential attachment graphs. We consider the push-pull strategy by Karp, Schindelhauer, Shenker, and Vocking [FOCS 2000], where in each round, each vertex chooses a random neighbor and exchanges information with it. We prove the following. The push-pull strategy delivers a message to all nodes within \Theta(\log n) rounds with high probability, where n is the number of nodes in the graph. The best known bound so far was O(\log^2 n) by Chierichetti, Lattanzi, and Panconesi [TCS 2011]. If we slightly modify the protocol so that contacts are chosen uniformly from all neighbors but the one contacted in the previous round, then this time reduces to \Theta(\logn/\log\log n). This is asymptotically optimal since it matches the diameter of the graph. In an asynchronous version of the protocol, the running time is shown to be even O(\sqrt{\log n}). In the second part, we consider the rumor spreading problem on the complete graph. We propose a new push protocol that achieves an asymptotically optimal time of (1+o(1))\log^2 n. It needs only O(n f(n)) calls, where f(n) = \omega(1) can be arbitrary. The protocol is robust against random node failures. We also extend it to deal with adversarial node failures efficiently. Im ersten Teil untersuchen wir die Verteilung von Informationen auf sozialen Netzwerken anhand des “Preferential Attachment” Modells. Hierzu betrachten wir das “Push-Pull” Protokoll von Karp, Schindelhauer, Shenker, and Vocking [FOCS 2000]: In jeder Runde wahlt ein Knoten einen zufalligen Nachbarknoten aus und tauscht sich mit ihm aus, d.h., wenn einer der beiden Knoten eine Information hat, erhalt sie der andere. Wir zeigen folgende Resultate. Das Push-Pull Protokoll verbreitet mit hoher Wahrscheinlichkeit eine Nachricht an alle Knoten innerhalb von \Theta(\log n) Runden, wobei n die Zahl der Knoten im Graph darstellt. Die beste bisher bekannte Laufzeitschranke war O(\log^2 n) von Chierichetti, Lattanzi, and Panconesi [TCS 2011]. Wenn wir das Protokoll leicht anpassen, so dass jeder Knoten bei der zufalligen Wahl eines Nachbarknoten den zuletzt kontaktierten ausschliest, verbessert sich diese Schranke auf \Theta(\log n/\log\log n). Dies ist asymptotisch optimal, da es dem Durchmesser des Graphen entspricht. In einer asynchronen Fassung des Protokolls reduziert sich die Laufzeit sogar auf O(\sqrt{\log n}). Im zweiten Teil betrachten wir die Verteilung von Informationen auf dem vollstandigen Graphen. Wir fuhren ein neues “Push” Protokoll ein, das eine asymptotisch optimale Laufzeit von (1+o(1))\log n erreicht. Dabei benotigt es nur O(n f(n)) Anrufe, wobei f(n) = \omega(1) beliebig ist. Das Protokoll ist zudem robust gegenuber zufalligen Knotenausfallen. Ferner erweitern wir das Protokoll, so dass es auch bei gezielten Knotenausfallen effizient bleibt.

We show that the asynchronous push-pull protocol spreads rumors in preferential attachment graphs (as defined by Barabási and Albert) in time $O(\sqrt{\log n})$ to all but a lower order fraction of the nodes with high probability. This is significantly faster than what synchronized protocols can achieve; an obvious lower bound for these is the average distance, which is known to be Θ(logn/loglogn).

A few hubs with many connections share with many individuals with few connections.

We provide a smoothed analysis of Hoare's find algorithm and we revisit the smoothed analysis of quicksort. Hoare's find algorithm - often called quickselect - is an easy-to-implement algorithm for finding the k-th smallest element of a sequence. While the worst-case number of comparisons that Hoare's find needs is quadratic, the average-case number is linear. We analyze what happens between these two extremes by providing a smoothed analysis of the algorithm in terms of two different perturbation models: additive noise and partial permutations. Moreover, we provide lower bounds for the smoothed number of comparisons of quicksort and Hoare's find for the median-of-three pivot rule, which usually yields faster algorithms than always selecting the first element: The pivot is the median of the first, middle, and last element of the sequence. We show that median-of-three does not yield a significant improvement over the classic rule: the lower bounds for the classic rule carry over to median-of-three.

Randomized rumor spreading was recently shown to be a very efficient mechanism to spread information in preferential attachment networks. Most interesting from the algorithm design point of view was the observation that the asymptotic run-time drops when memory is used to avoid re-contacting neighbors within a small number of rounds. In this experimental investigation, we confirm that a small amount of memory indeed reduces the run-time of the protocol even for small network sizes. We observe that one memory cell per node suffices to reduce the run-time significantly; more memory helps comparably little. Aside from extremely sparse graphs, preferential attachment graphs perform faster than all other graph classes examined. This holds independent of the amount of memory, but preferential attachment graphs benefit the most from the use of memory. We also analyze the influence of the network density and the size of the memory. For the asynchronous version of the rumor spreading protocol, we observe that the theoretically predicted asymptotic advantage of preferential attachment graphs is smaller than expected. There are other topologies which benefit even more from asynchrony. We complement our findings on artificial network models by the corresponding experiments on crawls of popular online social networks, where again we observe extremely rapid information dissemination and a sizable benefit from using memory and asynchrony.

The cover polynomial and its geometric version introduced by Chung & Graham and D’Antona & Munarini, respectively, are two-variate graph polynomials for directed graphs. They count the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, can be thought of as interpolations between determinant and permanent, and are proposed as directed analogues of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is # P -hard to evaluate at all but a few special points and curves. It turns out that the same holds for the cover polynomials: We prove that, in almost the whole plane, the problem of evaluating the cover polynomial and its geometric version is # P -hard under polynomial time Turing reductions, while only three points in the cover polynomial and two points in the geometric cover polynomial are easy. We also study the complexity of approximately evaluating the geometric cover polynomial. Under the reasonable complexity assumptions RP ≠ NP and RFP ≠ # P , we give a succinct characterization of a large class of points at which approximating the geometric cover polynomial within any polynomial factor is not possible.

Understanding structural and algorithmic properties of complex networks is an important task, not least because of the huge impact of the internet. Our focus is to analyze how news spreads in social networks. We simulate a simple information spreading process in different network topologies and demonstrate that news spreads much faster in existing social network topologies. We support this finding by analyzing information spreading in the mathematically defined preferential attachment network topology, which is a common model for real-world networks. We prove that here a sublogarithmic time suffices to spread a news to all nodes of the network. All previously studied network topologies need at least a logarithmic time. Surprisingly, we observe that nodes with few neighbors are crucial for the fast dissemination. Social networks like Facebook and Twitter are reshaping the way people take collective actions. They have played a crucial role in the recent uprisings of the ‘Arab Spring’ and the ‘London riots’. It has been argued that the ‘instantaneous nature’ of these networks influenced the speed at which the events were unfolding [4]. It is quite remarkable that social networks spread news so fast. Both the structure of social networks and the process that distributes the news are not designed with this purpose in mind. On the contrary, they are not designed at all, but have evolved in a random and decentralized manner. So is our view correct that social networks ease the spread of information (“rumors”), and if so, what particular properties of social networks are the reason for this? To answer these questions, we simulate a simple rumor spreading process on several graphs having the structure of existing large social networks. We see, for example, that a rumor started at a random node of the Twitter network in average reaches 45.6 million of the total of 51.2 million members within only eight rounds of communication. We also analyze this process on an abstract model of social networks, the so-called preferential attachment graphs introduced by Barabási and Albert [3]. In [17], we obtain a mathematical proof that rumors in such networks spread much faster than in many other network topologies—even faster than in networks having a communication link between any two nodes (complete graphs). As an explanation, we observe that nodes of small degree build a short-cut between those having large degree (hubs), which due to their large number of possible communication partners less often talk to each other directly.

With the prevalence of social networks, it has become increasingly important to understand their features and limitations. It has been observed that information spreads extremely fast in social networks. We study the performance of randomized rumor spreading protocols on graphs in the preferential attachment model. The well-known random phone call model of Karp et al. (FOCS 2000) is a push-pull strategy where in each round, each vertex chooses a random neighbor and exchanges information with it. We prove the following. - The push-pull strategy delivers a message to all nodes within Θ(log n) rounds with high probability. The best known bound so far was O(log2 n). - If we slightly modify the protocol so that contacts are chosen uniformly from all neighbors but the one contacted in the previous round, then this time reduces to Θ(log n / log log n), which is the diameter of the graph. This is the first time that a sublogarithmic broadcast time is proven for a natural setting. Also, this is the first time that avoiding double-contacts reduces the run-time to a smaller order of magnitude.

We propose a new protocol solving the fundamental problem of disseminating a piece of information to all members of a group of n players. It builds upon the classical randomized rumor spreading protocol and several extensions. The main achievements are the following: Our protocol spreads the rumor to all other nodes in the asymptotically optimal time of (1 + o(1)) \log_2 n. The whole process can be implemented in a way such that only O(n f(n)) calls are made, where f(n)= \omega(1) can be arbitrary. In contrast to other protocols suggested in the literature, our algorithm only uses push operations, i.e., only informed nodes take active actions in the network. To the best of our knowledge, this is the first randomized push algorithm that achieves an asymptotically optimal running time.

In this paper, we consider two-player zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a long-standing open question if a polynomial algorithm exists that solves BWR-games. In fact, a pseudo-polynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only two classes are known to have such a pseudo-polynomial algorithm: BW-games (the case with no random nodes) and ergodic BWR-games (in which the game's value does not depend on the initial position) with constant number of random nodes. In this paper, we show that the existence of a pseudo-polynomial algorithm for BWR-games with constant number of random vertices implies smoothed polynomial complexity and the existence of absolute and relative polynomial-time approximation schemes. In particular, we obtain smoothed polynomial complexity and derive absolute and relative approximation schemes for BW-games and ergodic BWR-games (assuming a technical requirement about the probabilities at the random nodes).

We introduce to the runtime analysis of evolutionary algorithms two powerful techniques: probability-generating functions and variable drift analysis. They are shown to provide a clean framework for proving sharp upper and lower bounds. As an application, we improve the results by Doerr et al. (GECCO~2010) in several respects. First, the upper bound on the expected running time of the most successful quasirandom evolutionary algorithm for the OneMax function is improved from 1.28n ln n to 0.982n ln n, which breaks the barrier of n ln n posed by coupon-collector processes. Compared to the classical 1+1-EA, whose runtime will for the first time be analyzed with respect to terms of lower order, this represents a speedup by more than a factor of e=2.71...

We give a time-randomness tradeoff for the quasi-random rumor spreading protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete graphs. In this protocol, the goal is to spread a piece of information originating from one vertex throughout the network. Each vertex is assumed to have a (cyclic) list of its neighbors. Once a vertex is informed by one of its neighbors, it chooses a position in its list uniformly at random and then informs its neighbors starting from that position and proceeding in order of the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron.~J.~Combin.~2009] showed that after $(1+o(1))(\log_2 n + \ln n)$ rounds, the rumor will have been broadcasted to all nodes with probability $1 - o(1)$. We study the broadcast time when the amount of randomness available at each node is reduced in natural way. In particular, we prove that if each node can only make its initial random selection from every $\ell$-th node on its list, then there exists lists such that $(1-\varepsilon) (\log_2 n + \ln n - \log_2 \ell - \ln \ell)+\ell-1$ steps are needed to inform every vertex with probability at least $1-O\bigl(\exp\bigl(-\frac{n^\varepsilon}{2\ln n}\bigr)\bigr)$. This shows that a further reduction of the amount of randomness used in a simple quasi-random protocol comes at a loss of efficiency.

We consider the following combinatorial auction: Given a range space , and m bidders interested in buying only ranges in , each bidder j declares her bid . We give a deterministic truthful mechanism, when the valuations are single-minded: when is a collection of fat objects (respectively, axis-aligned rectangles) in the plane, there is a truthful mechanism with a 1 + ε- (respectively, ⌈logn⌉)-approximation of the social welfare (where n is an upper bound on the maximum integral coordinate of each rectangle). We also consider the non-single-minded case, and design a randomized truthful-in-expectation mechanism with approximation guarantee O(1) (respectively, O(logm)).

We consider profit-maximization problems for combinatorial auctions with non-single minded valuation functions and limited supply. We obtain fairly general results that relate the approximability of the profit-maximization problem to that of the corresponding social-welfare-maximization (SWM) problem, which is the problem of finding an allocation ( S 1 , ..., S n ) satisfying the capacity constraints that has maximum total value ∑ j v j (S j ) . Our results apply to both structured valuation classes, such as subadditive valuations, as well as arbitrary valuations. For subadditive valuations (and hence submodular, XOS valuations), we obtain a solution with profit OPT SWM / O (log c max ), where OPT SWM is the optimum social welfare and c max is the maximum item-supply; thus, this yields an O (log c max )-approximation for the profit-maximization problem. Furthermore, given any class of valuation functions, if the SWM problem for this valuation class has an LP-relaxation (of a certain form) and an algorithm "verifying" an integrality gap of α for this LP, then we obtain a solution with profit OPT SWM / O (α log c max ), thus obtaining an O (α log c max )- approximation. The latter result implies an O (√ m log c max )-approximation for the profit maximization problem for combinatorial auctions with arbitrary valuations, and an O (log c max )-approximation for the non-single-minded tollbooth problem on trees. For the special case, when the tree is a path, we also obtain an incomparable O (log m )-approximation (via a different approach) for subadditive valuations, and arbitrary valuations with unlimited supply.1

Motivated by recent successful applications of the concept of quasirandomness, we investigate to what extent such ideas can be used in evolutionary computation. To this aim, we propose different variations of the classical (1+1) evolutionary algorithm, all imitating the property that the (1+1) EA over intervals of time touches all bits roughly the same number of times. We prove bounds on the optimization time of these algorithms for the simple OneMax function. Surprisingly, none of the algorithms achieves the seemingly obvious reduction of the runtime from Θ( n log n ) to O(n) . On the contrary, one may even need Ω( n 2 ) time. However, we also find that quasirandom ideas, if implemented correctly, can yield an over 50% speed-up.

Using the BBOB template, we investigate how the Nelder-Mead simplex algorithm can be combined with evolutionary ideas to give a competitive hybrid approach to optimize continuous functions. We significantly improve the performance of the algorithm in higher dimension by the addition of a reshaping step of the search, to correct for a known problem in the simplex search behaviour. We also give a reasonably good population-based approach in which only a third of the individuals is fully matured, with a bias towards fitter individuals, via a variant of the Nelder-Mead method.