Burkhard Monien

Ramsey numbers and an approximation algorithm for the vertex cover problem

SummaryWe show two results. First we derive an upper bound for the special Ramsey numbers rk(q) where rk(q) is the largest number of nodes a graph without odd cycles of length bounded by 2k+1 and without an independent set of size q+1 can have. We prove

Min cut is NP-complete for edge weighted trees

r_k (q) \leqq \frac{k}{{k + {\text{1}}}}q^{\frac{{k + {\text{1}}}}{k}} + \frac{{k + {\text{2}}}}{{k + {\text{1}}}}q

Nashification and the coordination ratio for a selfish routing game

The proof is constructive and yields an algorithm for computing an independent set of that size. Using this algorithm we secondly describe an O(¦V¦·¦E¦) time bounded approximation algorithm for the vertex cover problem, whose worst case ratio is

Diffusion Schemes for Load Balancing on Heterogeneous Networks

\Delta \leqq {\text{2 - }}\frac{{\text{1}}}{{k + {\text{1}}}}

Exact price of anarchy for polynomial congestion games

, for all graphs with at most (2k+3)k(2k+2) nodes (e.g. Δ≦1.8, if ¦V¦≦146000).

Selfish Routing with Incomplete Information

We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diffusion type. Within this framework we develop a new Optimal Polynomial Scheme (OPS) which we show to terminate within a finite number m of steps, where m only depends on the graph and not on the initial load distribution.We show that all existing diffusion load balancing algorithms, including OPS, determine a flow of load on the edges of the graph which is uniquely defined, independent of the method and minimal in the l2-norm. This result can also be extended to edge weighted graphs.The l2-minimality is achieved only if a diffusion algorithm is used as preprocessing and the real movement of load is performed in a second step. Thus, it is advisable to split the balancing process into the two steps of first determining a balancing flow and afterwards moving the load. We introduce the problem of scheduling a flow and present some first results on its complexity and the approximation quality of local greedy heuristics.