Extremal Optimization of Graph Partitioning at the Percolation Threshold

The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called Extremal Optimization, is compared to Simulated Annealing in extensive numerical simulations. While generally a complex (NP-hard) problem, the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold. At this threshold, the relative error of Simulated Annealing for large graphs is found to diverge relative to Extremal Optimization at equalized runtime. On the other hand, Extremal Optimization, based on the extremal dynamics of self-organized critical systems, reproduces known results about optimal partitions at this critical point quite well.Comment: 7 pages, RevTex, 9 ps-figures included, as to appear in Journal of Physics

Parameterized complexity of DPLL search procedures

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas

The Adversarial Stackelberg Value in Quantitative Games

In this paper, we study the notion of adversarial Stackelberg value for two-player non-zero sum games played on bi-weighted graphs with the mean-payoff and the discounted sum functions. The adversarial Stackelberg value of Player 0 is the largest value that Player 0 can obtain when announcing her strategy to Player 1 which in turn responds with any of his best response. For the mean-payoff function, we show that the adversarial Stackelberg value is not always achievable but ?-optimal strategies exist. We show how to compute this value and prove that the associated threshold problem is in NP. For the discounted sum payoff function, we draw a link with the target discounted sum problem which explains why the problem is difficult to solve for this payoff function. We also provide solutions to related gap problems

Letter graphs and geometric grid classes of permutations: characterization and recognition

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of graphs in this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author

Threshold phenomena in random graphs

In the 1950s, random graphs appeared for the first time in a result of the prolific hungarian mathematician Pál Erd\H{o}s. Since then, interest in random graph theory has only grown up until now. In its first stages, the basis of its theory were set, while they were mainly used in probability and combinatorics theory. However, with the new century and the boom of technologies like the World Wide Web, random graphs are even more important since they are extremely useful to handle problems in fields like network and communication theory. Because of this fact, nowadays random graphs are widely studied by the mathematical community around the world and new promising results have been recently achieved, showing an exciting future for this field. In this bachelor thesis, we focus our study on the threshold phenomena for graph properties within random graphs